The First, 10, moments of the Duration of a 3-Player Gambler's Ruin until one\ of them goes Broke By Shalosh B. Ekhad Suppose that 3 gamblers start with initial capitals of, A, B, C, dollars. At \ each round one of the players, picked uniformly at random, has to give o\ ne dollar to one of the other players (also uniformly at random). The game continues u\ ntil one of the players goes broke. Let X be duration of this game In this paper we will rederive, from scracth, Engel's expression for the exp\ ectation, and Bruss et. al.'s and Strizaker expression for the variance and state rigorously proved explicit expressions for the third all the way t\ o the, 10, -th moment. In particular we will give expressions for the ske\ wness and kurtosis. The Expectated Duration is: 3 A B C --------- A + B + C and in Maple notation 3*A*B*C/(A+B+C) Note that if N is large and the starting capitals are x1*N,x2*N,x3*N (where\ , of course, x1+x2+x3=1),, then the expectation is N^2 times 3 x1 x2 x3 ------------ x1 + x2 + x3 and in Maple notation 3*x1*x2*x3/(x1+x2+x3) The Second Moment is: 3 A B C (A B + A C + B C - 1) ----------------------------- 2 (A + B + C) and in Maple notation 3/2*A*B*C*(A*B+A*C+B*C-1)/(A+B+C) hence the variance (aka second moment about the mean), is: 2 2 2 2 2 2 3 A B C (A B + A C + A B - 3 A B C + A C + B C + B C - A - B - C) ----------------------------------------------------------------------- 2 2 (A + B + C) and in Maple notation 3/2*A*B*C*(A^2*B+A^2*C+A*B^2-3*A*B*C+A*C^2+B^2*C+B*C^2-A-B-C)/(A+B+C)^2 Note that if N is large and the starting capitals are x1*N,x2*N,x3*N, (where\ , of course, x1+x2+x3=1), then the limit of the variance divided by N^4,\ as N goes to infinity, is: 2 2 2 2 2 2 3 x1 x2 x3 (x1 x2 + x1 x3 + x1 x2 - 3 x1 x2 x3 + x1 x3 + x2 x3 + x2 x3 ) / 2 / (2 (x1 + x2 + x3) ) / and in Maple notation 3/2*x1*x2*x3*(x1^2*x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2*x3+x2*x3^2)/(x1+ x2+x3)^2 The , 3, -th (straight) moment of the Duration is: 3 3 2 2 2 2 2 3 2 3 A B C (3 A B + 3 A C + 12 A B + 24 A B C + 12 A C + 3 A B + 24 A B C 2 3 3 2 2 3 + 24 A B C + 3 A C + 3 B C + 12 B C + 3 B C - 30 A B - 30 A C - 30 B C + 4)/(40 (A + B + C)) and in Maple notation 3/40*A*B*C*(3*A^3*B+3*A^3*C+12*A^2*B^2+24*A^2*B*C+12*A^2*C^2+3*A*B^3+24*A*B^2*C +24*A*B*C^2+3*A*C^3+3*B^3*C+12*B^2*C^2+3*B*C^3-30*A*B-30*A*C-30*B*C+4)/(A+B+C) Hence the, 3, -th moment about the mean is 5 5 4 2 4 4 2 3 3 3 A B C (3 A B + 3 A C + 18 A B + 36 A B C + 18 A C + 30 A B 3 2 3 2 3 3 2 4 2 3 - 75 A B C - 75 A B C + 30 A C + 18 A B - 75 A B C 2 2 2 2 3 2 4 5 4 3 2 + 360 A B C - 75 A B C + 18 A C + 3 A B + 36 A B C - 75 A B C 2 3 4 5 5 4 2 3 3 - 75 A B C + 36 A B C + 3 A C + 3 B C + 18 B C + 30 B C 2 4 5 3 3 2 2 2 2 2 + 18 B C + 3 B C - 30 A B - 30 A C - 60 A B + 30 A B C - 60 A C 3 2 2 3 3 2 2 3 - 30 A B + 30 A B C + 30 A B C - 30 A C - 30 B C - 60 B C - 30 B C 2 2 2 / 3 + 4 A + 8 A B + 8 A C + 4 B + 8 B C + 4 C ) / (40 (A + B + C) ) / and in Maple notation 3/40*A*B*C*(3*A^5*B+3*A^5*C+18*A^4*B^2+36*A^4*B*C+18*A^4*C^2+30*A^3*B^3-75*A^3* B^2*C-75*A^3*B*C^2+30*A^3*C^3+18*A^2*B^4-75*A^2*B^3*C+360*A^2*B^2*C^2-75*A^2*B* C^3+18*A^2*C^4+3*A*B^5+36*A*B^4*C-75*A*B^3*C^2-75*A*B^2*C^3+36*A*B*C^4+3*A*C^5+ 3*B^5*C+18*B^4*C^2+30*B^3*C^3+18*B^2*C^4+3*B*C^5-30*A^3*B-30*A^3*C-60*A^2*B^2+ 30*A^2*B*C-60*A^2*C^2-30*A*B^3+30*A*B^2*C+30*A*B*C^2-30*A*C^3-30*B^3*C-60*B^2*C ^2-30*B*C^3+4*A^2+8*A*B+8*A*C+4*B^2+8*B*C+4*C^2)/(A+B+C)^3 Hence the scaled, 3, -th moment about the mean is aka skewness 5 5 4 2 4 4 2 3 3 A B C (3 A B + 3 A C + 18 A B + 36 A B C + 18 A C + 30 A B 3 2 3 2 3 3 2 4 2 3 - 75 A B C - 75 A B C + 30 A C + 18 A B - 75 A B C 2 2 2 2 3 2 4 5 4 3 2 + 360 A B C - 75 A B C + 18 A C + 3 A B + 36 A B C - 75 A B C 2 3 4 5 5 4 2 3 3 - 75 A B C + 36 A B C + 3 A C + 3 B C + 18 B C + 30 B C 2 4 5 3 3 2 2 2 2 2 + 18 B C + 3 B C - 30 A B - 30 A C - 60 A B + 30 A B C - 60 A C 3 2 2 3 3 2 2 3 - 30 A B + 30 A B C + 30 A B C - 30 A C - 30 B C - 60 B C - 30 B C / 2 2 2 1/2 1/2 / | + 4 A + 8 A B + 8 A C + 4 B + 8 B C + 4 C ) 3 2 / |60 / | \ 3 (A + B + C) / 2 2 2 2 2 2 \3/2\ |A B C (A B + A C + A B - 3 A B C + A C + B C + B C - A - B - C)| | |---------------------------------------------------------------------| | | 2 | | \ (A + B + C) / / and in Maple notation 1/60*A*B*C*(3*A^5*B+3*A^5*C+18*A^4*B^2+36*A^4*B*C+18*A^4*C^2+30*A^3*B^3-75*A^3* B^2*C-75*A^3*B*C^2+30*A^3*C^3+18*A^2*B^4-75*A^2*B^3*C+360*A^2*B^2*C^2-75*A^2*B* C^3+18*A^2*C^4+3*A*B^5+36*A*B^4*C-75*A*B^3*C^2-75*A*B^2*C^3+36*A*B*C^4+3*A*C^5+ 3*B^5*C+18*B^4*C^2+30*B^3*C^3+18*B^2*C^4+3*B*C^5-30*A^3*B-30*A^3*C-60*A^2*B^2+ 30*A^2*B*C-60*A^2*C^2-30*A*B^3+30*A*B^2*C+30*A*B*C^2-30*A*C^3-30*B^3*C-60*B^2*C ^2-30*B*C^3+4*A^2+8*A*B+8*A*C+4*B^2+8*B*C+4*C^2)/(A+B+C)^3*3^(1/2)*2^(1/2)/(A*B *C*(A^2*B+A^2*C+A*B^2-3*A*B*C+A*C^2+B^2*C+B*C^2-A-B-C)/(A+B+C)^2)^(3/2) Note that if N is large and the starting capitals are x1*N,x2*N,x3*N, (where\ , of course, x1+x2+x3=1), then the limit of the, 3, -th scaled moment about the mean tends to 6 2 1/2 1/2 6 2 1/2 1/2 5 3 1/2 1/2 (x1 x2 x3 3 2 + x1 x2 x3 3 2 + 6 x1 x2 x3 3 2 5 2 2 1/2 1/2 5 3 1/2 1/2 + 12 x1 x2 x3 3 2 + 6 x1 x2 x3 3 2 4 4 1/2 1/2 4 3 2 1/2 1/2 + 10 x1 x2 x3 3 2 - 25 x1 x2 x3 3 2 4 2 3 1/2 1/2 4 4 1/2 1/2 - 25 x1 x2 x3 3 2 + 10 x1 x2 x3 3 2 3 5 1/2 1/2 3 4 2 1/2 1/2 + 6 x1 x2 x3 3 2 - 25 x1 x2 x3 3 2 3 3 3 1/2 1/2 3 2 4 1/2 1/2 + 120 x1 x2 x3 3 2 - 25 x1 x2 x3 3 2 3 5 1/2 1/2 2 6 1/2 1/2 2 5 2 1/2 1/2 + 6 x1 x2 x3 3 2 + x1 x2 x3 3 2 + 12 x1 x2 x3 3 2 2 4 3 1/2 1/2 2 3 4 1/2 1/2 - 25 x1 x2 x3 3 2 - 25 x1 x2 x3 3 2 2 2 5 1/2 1/2 2 6 1/2 1/2 6 2 1/2 1/2 + 12 x1 x2 x3 3 2 + x1 x2 x3 3 2 + x1 x2 x3 3 2 5 3 1/2 1/2 4 4 1/2 1/2 + 6 x1 x2 x3 3 2 + 10 x1 x2 x3 3 2 3 5 1/2 1/2 2 6 1/2 1/2 / 3 + 6 x1 x2 x3 3 2 + x1 x2 x3 3 2 ) / (20 %1 x1 / 2 2 2 + 60 %1 x1 x2 + 60 %1 x1 x3 + 60 %1 x1 x2 + 120 %1 x3 x2 x1 2 3 2 2 3 + 60 %1 x1 x3 + 20 %1 x2 + 60 %1 x2 x3 + 60 %1 x2 x3 + 20 %1 x3 ) / %1 := |x1 x2 x3 \ 2 2 2 2 2 2 / (x1 x2 + x1 x3 + x1 x2 - 3 x1 x2 x3 + x1 x3 + x2 x3 + x2 x3 ) / ( / 2 2 2 \(3/2) x1 + 2 x1 x2 + 2 x1 x3 + x2 + 2 x2 x3 + x3 )| / and in Maple notation (x1^6*x2^2*x3*3^(1/2)*2^(1/2)+x1^6*x2*x3^2*3^(1/2)*2^(1/2)+6*x1^5*x2^3*x3*3^(1/ 2)*2^(1/2)+12*x1^5*x2^2*x3^2*3^(1/2)*2^(1/2)+6*x1^5*x2*x3^3*3^(1/2)*2^(1/2)+10* x1^4*x2^4*x3*3^(1/2)*2^(1/2)-25*x1^4*x2^3*x3^2*3^(1/2)*2^(1/2)-25*x1^4*x2^2*x3^ 3*3^(1/2)*2^(1/2)+10*x1^4*x2*x3^4*3^(1/2)*2^(1/2)+6*x1^3*x2^5*x3*3^(1/2)*2^(1/2 )-25*x1^3*x2^4*x3^2*3^(1/2)*2^(1/2)+120*x1^3*x2^3*x3^3*3^(1/2)*2^(1/2)-25*x1^3* x2^2*x3^4*3^(1/2)*2^(1/2)+6*x1^3*x2*x3^5*3^(1/2)*2^(1/2)+x1^2*x2^6*x3*3^(1/2)*2 ^(1/2)+12*x1^2*x2^5*x3^2*3^(1/2)*2^(1/2)-25*x1^2*x2^4*x3^3*3^(1/2)*2^(1/2)-25* x1^2*x2^3*x3^4*3^(1/2)*2^(1/2)+12*x1^2*x2^2*x3^5*3^(1/2)*2^(1/2)+x1^2*x2*x3^6*3 ^(1/2)*2^(1/2)+x1*x2^6*x3^2*3^(1/2)*2^(1/2)+6*x1*x2^5*x3^3*3^(1/2)*2^(1/2)+10* x1*x2^4*x3^4*3^(1/2)*2^(1/2)+6*x1*x2^3*x3^5*3^(1/2)*2^(1/2)+x1*x2^2*x3^6*3^(1/2 )*2^(1/2))/(20*(x1*x2*x3*(x1^2*x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2*x3+x2 *x3^2)/(x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2*x3+x3^2))^(3/2)*x1^3+60*(x1*x2*x3*(x1^2* x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2*x3+x2*x3^2)/(x1^2+2*x1*x2+2*x1*x3+x2 ^2+2*x2*x3+x3^2))^(3/2)*x1^2*x2+60*(x1*x2*x3*(x1^2*x2+x1^2*x3+x1*x2^2-3*x1*x2* x3+x1*x3^2+x2^2*x3+x2*x3^2)/(x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2*x3+x3^2))^(3/2)*x1^ 2*x3+60*(x1*x2*x3*(x1^2*x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2*x3+x2*x3^2)/ (x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2*x3+x3^2))^(3/2)*x1*x2^2+120*(x1*x2*x3*(x1^2*x2+ x1^2*x3+x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2*x3+x2*x3^2)/(x1^2+2*x1*x2+2*x1*x3+x2^2+ 2*x2*x3+x3^2))^(3/2)*x3*x2*x1+60*(x1*x2*x3*(x1^2*x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+ x1*x3^2+x2^2*x3+x2*x3^2)/(x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2*x3+x3^2))^(3/2)*x1*x3^ 2+20*(x1*x2*x3*(x1^2*x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2*x3+x2*x3^2)/(x1 ^2+2*x1*x2+2*x1*x3+x2^2+2*x2*x3+x3^2))^(3/2)*x2^3+60*(x1*x2*x3*(x1^2*x2+x1^2*x3 +x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2*x3+x2*x3^2)/(x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2*x3 +x3^2))^(3/2)*x2^2*x3+60*(x1*x2*x3*(x1^2*x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+x1*x3^2+ x2^2*x3+x2*x3^2)/(x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2*x3+x3^2))^(3/2)*x2*x3^2+20*(x1 *x2*x3*(x1^2*x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2*x3+x2*x3^2)/(x1^2+2*x1* x2+2*x1*x3+x2^2+2*x2*x3+x3^2))^(3/2)*x3^3) The , 4, -th (straight) moment of the Duration is: 5 5 4 2 4 4 2 3 3 3 A B C (6 A B + 6 A C + 36 A B + 72 A B C + 36 A C + 69 A B 3 2 3 2 3 3 2 4 2 3 + 228 A B C + 228 A B C + 69 A C + 36 A B + 228 A B C 2 2 2 2 3 2 4 5 4 3 2 + 384 A B C + 228 A B C + 36 A C + 6 A B + 72 A B C + 228 A B C 2 3 4 5 5 4 2 3 3 + 228 A B C + 72 A B C + 6 A C + 6 B C + 36 B C + 69 B C 2 4 5 3 3 2 2 2 + 36 B C + 6 B C - 63 A B - 63 A C - 252 A B - 504 A B C 2 2 3 2 2 3 3 - 252 A C - 63 A B - 504 A B C - 504 A B C - 63 A C - 63 B C 2 2 3 - 252 B C - 63 B C + 161 A B + 161 A C + 161 B C + 16)/(280 (A + B + C) ) and in Maple notation 3/280*A*B*C*(6*A^5*B+6*A^5*C+36*A^4*B^2+72*A^4*B*C+36*A^4*C^2+69*A^3*B^3+228*A^ 3*B^2*C+228*A^3*B*C^2+69*A^3*C^3+36*A^2*B^4+228*A^2*B^3*C+384*A^2*B^2*C^2+228*A ^2*B*C^3+36*A^2*C^4+6*A*B^5+72*A*B^4*C+228*A*B^3*C^2+228*A*B^2*C^3+72*A*B*C^4+6 *A*C^5+6*B^5*C+36*B^4*C^2+69*B^3*C^3+36*B^2*C^4+6*B*C^5-63*A^3*B-63*A^3*C-252*A ^2*B^2-504*A^2*B*C-252*A^2*C^2-63*A*B^3-504*A*B^2*C-504*A*B*C^2-63*A*C^3-63*B^3 *C-252*B^2*C^2-63*B*C^3+161*A*B+161*A*C+161*B*C+16)/(A+B+C) Hence the, 4, -th moment about the mean is 8 8 7 2 7 7 2 6 3 3 A B C (6 A B + 6 A C + 54 A B + 108 A B C + 54 A C + 195 A B 6 2 6 2 6 3 5 4 5 3 + 354 A B C + 354 A B C + 195 A C + 357 A B + 63 A B C 5 2 2 5 3 5 4 4 5 4 4 - 588 A B C + 63 A B C + 357 A C + 357 A B - 378 A B C 4 3 2 4 2 3 4 4 4 5 3 6 + 3423 A B C + 3423 A B C - 378 A B C + 357 A C + 195 A B 3 5 3 4 2 3 3 3 3 2 4 3 5 + 63 A B C + 3423 A B C - 8505 A B C + 3423 A B C + 63 A B C 3 6 2 7 2 6 2 5 2 2 4 3 + 195 A C + 54 A B + 354 A B C - 588 A B C + 3423 A B C 2 3 4 2 2 5 2 6 2 7 8 + 3423 A B C - 588 A B C + 354 A B C + 54 A C + 6 A B 7 6 2 5 3 4 4 3 5 + 108 A B C + 354 A B C + 63 A B C - 378 A B C + 63 A B C 2 6 7 8 8 7 2 6 3 + 354 A B C + 108 A B C + 6 A C + 6 B C + 54 B C + 195 B C 5 4 4 5 3 6 2 7 8 6 + 357 B C + 357 B C + 195 B C + 54 B C + 6 B C - 63 A B 6 5 2 5 5 2 4 3 4 2 - 63 A C - 441 A B - 882 A B C - 441 A C - 1008 A B - 819 A B C 4 2 4 3 3 4 3 2 2 3 4 - 819 A B C - 1008 A C - 1008 A B - 3150 A B C - 1008 A C 2 5 2 4 2 3 2 2 2 3 2 4 - 441 A B - 819 A B C - 3150 A B C - 3150 A B C - 819 A B C 2 5 6 5 4 2 2 4 - 441 A C - 63 A B - 882 A B C - 819 A B C - 819 A B C 5 6 6 5 2 4 3 3 4 - 882 A B C - 63 A C - 63 B C - 441 B C - 1008 B C - 1008 B C 2 5 6 4 4 3 2 3 - 441 B C - 63 B C + 161 A B + 161 A C + 483 A B + 791 A B C 3 2 2 3 2 2 2 2 2 3 + 483 A C + 483 A B + 1260 A B C + 1260 A B C + 483 A C 4 3 2 2 3 4 4 + 161 A B + 791 A B C + 1260 A B C + 791 A B C + 161 A C + 161 B C 3 2 2 3 4 3 2 2 2 + 483 B C + 483 B C + 161 B C + 16 A + 48 A B + 48 A C + 48 A B 2 3 2 2 3 / + 96 A B C + 48 A C + 16 B + 48 B C + 48 B C + 16 C ) / (280 / 4 (A + B + C) ) and in Maple notation 3/280*A*B*C*(6*A^8*B+6*A^8*C+54*A^7*B^2+108*A^7*B*C+54*A^7*C^2+195*A^6*B^3+354* A^6*B^2*C+354*A^6*B*C^2+195*A^6*C^3+357*A^5*B^4+63*A^5*B^3*C-588*A^5*B^2*C^2+63 *A^5*B*C^3+357*A^5*C^4+357*A^4*B^5-378*A^4*B^4*C+3423*A^4*B^3*C^2+3423*A^4*B^2* C^3-378*A^4*B*C^4+357*A^4*C^5+195*A^3*B^6+63*A^3*B^5*C+3423*A^3*B^4*C^2-8505*A^ 3*B^3*C^3+3423*A^3*B^2*C^4+63*A^3*B*C^5+195*A^3*C^6+54*A^2*B^7+354*A^2*B^6*C-\ 588*A^2*B^5*C^2+3423*A^2*B^4*C^3+3423*A^2*B^3*C^4-588*A^2*B^2*C^5+354*A^2*B*C^6 +54*A^2*C^7+6*A*B^8+108*A*B^7*C+354*A*B^6*C^2+63*A*B^5*C^3-378*A*B^4*C^4+63*A*B ^3*C^5+354*A*B^2*C^6+108*A*B*C^7+6*A*C^8+6*B^8*C+54*B^7*C^2+195*B^6*C^3+357*B^5 *C^4+357*B^4*C^5+195*B^3*C^6+54*B^2*C^7+6*B*C^8-63*A^6*B-63*A^6*C-441*A^5*B^2-\ 882*A^5*B*C-441*A^5*C^2-1008*A^4*B^3-819*A^4*B^2*C-819*A^4*B*C^2-1008*A^4*C^3-\ 1008*A^3*B^4-3150*A^3*B^2*C^2-1008*A^3*C^4-441*A^2*B^5-819*A^2*B^4*C-3150*A^2*B ^3*C^2-3150*A^2*B^2*C^3-819*A^2*B*C^4-441*A^2*C^5-63*A*B^6-882*A*B^5*C-819*A*B^ 4*C^2-819*A*B^2*C^4-882*A*B*C^5-63*A*C^6-63*B^6*C-441*B^5*C^2-1008*B^4*C^3-1008 *B^3*C^4-441*B^2*C^5-63*B*C^6+161*A^4*B+161*A^4*C+483*A^3*B^2+791*A^3*B*C+483*A ^3*C^2+483*A^2*B^3+1260*A^2*B^2*C+1260*A^2*B*C^2+483*A^2*C^3+161*A*B^4+791*A*B^ 3*C+1260*A*B^2*C^2+791*A*B*C^3+161*A*C^4+161*B^4*C+483*B^3*C^2+483*B^2*C^3+161* B*C^4+16*A^3+48*A^2*B+48*A^2*C+48*A*B^2+96*A*B*C+48*A*C^2+16*B^3+48*B^2*C+48*B* C^2+16*C^3)/(A+B+C)^4 Hence the scaled, 4, -th moment about the mean is aka kurtosis 8 8 7 2 7 7 2 6 3 6 2 (6 A B + 6 A C + 54 A B + 108 A B C + 54 A C + 195 A B + 354 A B C 6 2 6 3 5 4 5 3 5 2 2 + 354 A B C + 195 A C + 357 A B + 63 A B C - 588 A B C 5 3 5 4 4 5 4 4 4 3 2 + 63 A B C + 357 A C + 357 A B - 378 A B C + 3423 A B C 4 2 3 4 4 4 5 3 6 3 5 + 3423 A B C - 378 A B C + 357 A C + 195 A B + 63 A B C 3 4 2 3 3 3 3 2 4 3 5 3 6 + 3423 A B C - 8505 A B C + 3423 A B C + 63 A B C + 195 A C 2 7 2 6 2 5 2 2 4 3 2 3 4 + 54 A B + 354 A B C - 588 A B C + 3423 A B C + 3423 A B C 2 2 5 2 6 2 7 8 7 - 588 A B C + 354 A B C + 54 A C + 6 A B + 108 A B C 6 2 5 3 4 4 3 5 2 6 + 354 A B C + 63 A B C - 378 A B C + 63 A B C + 354 A B C 7 8 8 7 2 6 3 5 4 + 108 A B C + 6 A C + 6 B C + 54 B C + 195 B C + 357 B C 4 5 3 6 2 7 8 6 6 + 357 B C + 195 B C + 54 B C + 6 B C - 63 A B - 63 A C 5 2 5 5 2 4 3 4 2 - 441 A B - 882 A B C - 441 A C - 1008 A B - 819 A B C 4 2 4 3 3 4 3 2 2 3 4 - 819 A B C - 1008 A C - 1008 A B - 3150 A B C - 1008 A C 2 5 2 4 2 3 2 2 2 3 2 4 - 441 A B - 819 A B C - 3150 A B C - 3150 A B C - 819 A B C 2 5 6 5 4 2 2 4 - 441 A C - 63 A B - 882 A B C - 819 A B C - 819 A B C 5 6 6 5 2 4 3 3 4 - 882 A B C - 63 A C - 63 B C - 441 B C - 1008 B C - 1008 B C 2 5 6 4 4 3 2 3 - 441 B C - 63 B C + 161 A B + 161 A C + 483 A B + 791 A B C 3 2 2 3 2 2 2 2 2 3 + 483 A C + 483 A B + 1260 A B C + 1260 A B C + 483 A C 4 3 2 2 3 4 4 + 161 A B + 791 A B C + 1260 A B C + 791 A B C + 161 A C + 161 B C 3 2 2 3 4 3 2 2 2 + 483 B C + 483 B C + 161 B C + 16 A + 48 A B + 48 A C + 48 A B 2 3 2 2 3 / + 96 A B C + 48 A C + 16 B + 48 B C + 48 B C + 16 C ) / (210 A B C / 2 2 2 2 2 2 2 (A B + A C + A B - 3 A B C + A C + B C + B C - A - B - C) ) and in Maple notation 1/210/A/B/C*(6*A^8*B+6*A^8*C+54*A^7*B^2+108*A^7*B*C+54*A^7*C^2+195*A^6*B^3+354* A^6*B^2*C+354*A^6*B*C^2+195*A^6*C^3+357*A^5*B^4+63*A^5*B^3*C-588*A^5*B^2*C^2+63 *A^5*B*C^3+357*A^5*C^4+357*A^4*B^5-378*A^4*B^4*C+3423*A^4*B^3*C^2+3423*A^4*B^2* C^3-378*A^4*B*C^4+357*A^4*C^5+195*A^3*B^6+63*A^3*B^5*C+3423*A^3*B^4*C^2-8505*A^ 3*B^3*C^3+3423*A^3*B^2*C^4+63*A^3*B*C^5+195*A^3*C^6+54*A^2*B^7+354*A^2*B^6*C-\ 588*A^2*B^5*C^2+3423*A^2*B^4*C^3+3423*A^2*B^3*C^4-588*A^2*B^2*C^5+354*A^2*B*C^6 +54*A^2*C^7+6*A*B^8+108*A*B^7*C+354*A*B^6*C^2+63*A*B^5*C^3-378*A*B^4*C^4+63*A*B ^3*C^5+354*A*B^2*C^6+108*A*B*C^7+6*A*C^8+6*B^8*C+54*B^7*C^2+195*B^6*C^3+357*B^5 *C^4+357*B^4*C^5+195*B^3*C^6+54*B^2*C^7+6*B*C^8-63*A^6*B-63*A^6*C-441*A^5*B^2-\ 882*A^5*B*C-441*A^5*C^2-1008*A^4*B^3-819*A^4*B^2*C-819*A^4*B*C^2-1008*A^4*C^3-\ 1008*A^3*B^4-3150*A^3*B^2*C^2-1008*A^3*C^4-441*A^2*B^5-819*A^2*B^4*C-3150*A^2*B ^3*C^2-3150*A^2*B^2*C^3-819*A^2*B*C^4-441*A^2*C^5-63*A*B^6-882*A*B^5*C-819*A*B^ 4*C^2-819*A*B^2*C^4-882*A*B*C^5-63*A*C^6-63*B^6*C-441*B^5*C^2-1008*B^4*C^3-1008 *B^3*C^4-441*B^2*C^5-63*B*C^6+161*A^4*B+161*A^4*C+483*A^3*B^2+791*A^3*B*C+483*A ^3*C^2+483*A^2*B^3+1260*A^2*B^2*C+1260*A^2*B*C^2+483*A^2*C^3+161*A*B^4+791*A*B^ 3*C+1260*A*B^2*C^2+791*A*B*C^3+161*A*C^4+161*B^4*C+483*B^3*C^2+483*B^2*C^3+161* B*C^4+16*A^3+48*A^2*B+48*A^2*C+48*A*B^2+96*A*B*C+48*A*C^2+16*B^3+48*B^2*C+48*B* C^2+16*C^3)/(A^2*B+A^2*C+A*B^2-3*A*B*C+A*C^2+B^2*C+B*C^2-A-B-C)^2 Note that if N is large and the starting capitals are x1*N,x2*N,x3*N, (where\ , of course, x1+x2+x3=1), then the limit of the, 4, -th scaled moment about the mean tends to 8 8 7 2 7 7 2 6 3 (2 x1 x2 + 2 x1 x3 + 18 x1 x2 + 36 x1 x2 x3 + 18 x1 x3 + 65 x1 x2 6 2 6 2 6 3 5 4 + 118 x1 x2 x3 + 118 x1 x2 x3 + 65 x1 x3 + 119 x1 x2 5 3 5 2 2 5 3 5 4 + 21 x1 x2 x3 - 196 x1 x2 x3 + 21 x1 x2 x3 + 119 x1 x3 4 5 4 4 4 3 2 4 2 3 + 119 x1 x2 - 126 x1 x2 x3 + 1141 x1 x2 x3 + 1141 x1 x2 x3 4 4 4 5 3 6 3 5 - 126 x1 x2 x3 + 119 x1 x3 + 65 x1 x2 + 21 x1 x2 x3 3 4 2 3 3 3 3 2 4 3 5 + 1141 x1 x2 x3 - 2835 x1 x2 x3 + 1141 x1 x2 x3 + 21 x1 x2 x3 3 6 2 7 2 6 2 5 2 + 65 x1 x3 + 18 x1 x2 + 118 x1 x2 x3 - 196 x1 x2 x3 2 4 3 2 3 4 2 2 5 2 6 + 1141 x1 x2 x3 + 1141 x1 x2 x3 - 196 x1 x2 x3 + 118 x1 x2 x3 2 7 8 7 6 2 5 3 + 18 x1 x3 + 2 x1 x2 + 36 x1 x2 x3 + 118 x1 x2 x3 + 21 x1 x2 x3 4 4 3 5 2 6 7 - 126 x1 x2 x3 + 21 x1 x2 x3 + 118 x1 x2 x3 + 36 x1 x2 x3 8 8 7 2 6 3 5 4 + 2 x1 x3 + 2 x2 x3 + 18 x2 x3 + 65 x2 x3 + 119 x2 x3 4 5 3 6 2 7 8 / + 119 x2 x3 + 65 x2 x3 + 18 x2 x3 + 2 x2 x3 ) / (70 x1 x2 x3 / 2 2 2 2 2 2 2 (x1 x2 + x1 x3 + x1 x2 - 3 x1 x2 x3 + x1 x3 + x2 x3 + x2 x3 ) ) and in Maple notation 1/70*(2*x1^8*x2+2*x1^8*x3+18*x1^7*x2^2+36*x1^7*x2*x3+18*x1^7*x3^2+65*x1^6*x2^3+ 118*x1^6*x2^2*x3+118*x1^6*x2*x3^2+65*x1^6*x3^3+119*x1^5*x2^4+21*x1^5*x2^3*x3-\ 196*x1^5*x2^2*x3^2+21*x1^5*x2*x3^3+119*x1^5*x3^4+119*x1^4*x2^5-126*x1^4*x2^4*x3 +1141*x1^4*x2^3*x3^2+1141*x1^4*x2^2*x3^3-126*x1^4*x2*x3^4+119*x1^4*x3^5+65*x1^3 *x2^6+21*x1^3*x2^5*x3+1141*x1^3*x2^4*x3^2-2835*x1^3*x2^3*x3^3+1141*x1^3*x2^2*x3 ^4+21*x1^3*x2*x3^5+65*x1^3*x3^6+18*x1^2*x2^7+118*x1^2*x2^6*x3-196*x1^2*x2^5*x3^ 2+1141*x1^2*x2^4*x3^3+1141*x1^2*x2^3*x3^4-196*x1^2*x2^2*x3^5+118*x1^2*x2*x3^6+ 18*x1^2*x3^7+2*x1*x2^8+36*x1*x2^7*x3+118*x1*x2^6*x3^2+21*x1*x2^5*x3^3-126*x1*x2 ^4*x3^4+21*x1*x2^3*x3^5+118*x1*x2^2*x3^6+36*x1*x2*x3^7+2*x1*x3^8+2*x2^8*x3+18* x2^7*x3^2+65*x2^6*x3^3+119*x2^5*x3^4+119*x2^4*x3^5+65*x2^3*x3^6+18*x2^2*x3^7+2* x2*x3^8)/x1/x2/x3/(x1^2*x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2*x3+x2*x3^2)^ 2 The , 5, -th (straight) moment of the Duration is: 7 7 6 2 6 6 2 5 3 3 A B C (9 A B + 9 A C + 72 A B + 144 A B C + 72 A C + 222 A B 5 2 5 2 5 3 4 4 4 3 + 696 A B C + 696 A B C + 222 A C + 324 A B + 1476 A B C 4 2 2 4 3 4 4 3 5 3 4 + 2304 A B C + 1476 A B C + 324 A C + 222 A B + 1476 A B C 3 3 2 3 2 3 3 4 3 5 2 6 + 3360 A B C + 3360 A B C + 1476 A B C + 222 A C + 72 A B 2 5 2 4 2 2 3 3 2 2 4 + 696 A B C + 2304 A B C + 3360 A B C + 2304 A B C 2 5 2 6 7 6 5 2 + 696 A B C + 72 A C + 9 A B + 144 A B C + 696 A B C 4 3 3 4 2 5 6 7 7 + 1476 A B C + 1476 A B C + 696 A B C + 144 A B C + 9 A C + 9 B C 6 2 5 3 4 4 3 5 2 6 7 + 72 B C + 222 B C + 324 B C + 222 B C + 72 B C + 9 B C 5 5 4 2 4 4 2 3 3 - 120 A B - 120 A C - 720 A B - 1440 A B C - 720 A C - 1380 A B 3 2 3 2 3 3 2 4 2 3 - 4560 A B C - 4560 A B C - 1380 A C - 720 A B - 4560 A B C 2 2 2 2 3 2 4 5 4 - 7680 A B C - 4560 A B C - 720 A C - 120 A B - 1440 A B C 3 2 2 3 4 5 5 - 4560 A B C - 4560 A B C - 1440 A B C - 120 A C - 120 B C 4 2 3 3 2 4 5 3 3 - 720 B C - 1380 B C - 720 B C - 120 B C + 399 A B + 399 A C 2 2 2 2 2 3 2 + 1596 A B + 3192 A B C + 1596 A C + 399 A B + 3192 A B C 2 3 3 2 2 3 + 3192 A B C + 399 A C + 399 B C + 1596 B C + 399 B C - 120 A B - 120 A C - 120 B C + 16)/(1120 (A + B + C)) and in Maple notation 3/1120*A*B*C*(9*A^7*B+9*A^7*C+72*A^6*B^2+144*A^6*B*C+72*A^6*C^2+222*A^5*B^3+696 *A^5*B^2*C+696*A^5*B*C^2+222*A^5*C^3+324*A^4*B^4+1476*A^4*B^3*C+2304*A^4*B^2*C^ 2+1476*A^4*B*C^3+324*A^4*C^4+222*A^3*B^5+1476*A^3*B^4*C+3360*A^3*B^3*C^2+3360*A ^3*B^2*C^3+1476*A^3*B*C^4+222*A^3*C^5+72*A^2*B^6+696*A^2*B^5*C+2304*A^2*B^4*C^2 +3360*A^2*B^3*C^3+2304*A^2*B^2*C^4+696*A^2*B*C^5+72*A^2*C^6+9*A*B^7+144*A*B^6*C +696*A*B^5*C^2+1476*A*B^4*C^3+1476*A*B^3*C^4+696*A*B^2*C^5+144*A*B*C^6+9*A*C^7+ 9*B^7*C+72*B^6*C^2+222*B^5*C^3+324*B^4*C^4+222*B^3*C^5+72*B^2*C^6+9*B*C^7-120*A ^5*B-120*A^5*C-720*A^4*B^2-1440*A^4*B*C-720*A^4*C^2-1380*A^3*B^3-4560*A^3*B^2*C -4560*A^3*B*C^2-1380*A^3*C^3-720*A^2*B^4-4560*A^2*B^3*C-7680*A^2*B^2*C^2-4560*A ^2*B*C^3-720*A^2*C^4-120*A*B^5-1440*A*B^4*C-4560*A*B^3*C^2-4560*A*B^2*C^3-1440* A*B*C^4-120*A*C^5-120*B^5*C-720*B^4*C^2-1380*B^3*C^3-720*B^2*C^4-120*B*C^5+399* A^3*B+399*A^3*C+1596*A^2*B^2+3192*A^2*B*C+1596*A^2*C^2+399*A*B^3+3192*A*B^2*C+ 3192*A*B*C^2+399*A*C^3+399*B^3*C+1596*B^2*C^2+399*B*C^3-120*A*B-120*A*C-120*B*C +16)/(A+B+C) Hence the, 5, -th moment about the mean is 11 11 10 2 10 10 2 9 3 3 A B C (9 A B + 9 A C + 108 A B + 216 A B C + 108 A C + 564 A B 9 2 9 2 9 3 8 4 8 3 + 1362 A B C + 1362 A B C + 564 A C + 1680 A B + 3780 A B C 8 2 2 8 3 8 4 7 5 7 4 + 4200 A B C + 3780 A B C + 1680 A C + 3147 A B + 5301 A B C 7 3 2 7 2 3 7 4 7 5 6 6 + 6510 A B C + 6510 A B C + 5301 A B C + 3147 A C + 3864 A B 6 5 6 4 2 6 3 3 6 2 4 + 4692 A B C + 20868 A B C + 40080 A B C + 20868 A B C 6 5 6 6 5 7 5 6 5 5 2 + 4692 A B C + 3864 A C + 3147 A B + 4692 A B C + 34608 A B C 5 4 3 5 3 4 5 2 5 5 6 - 13482 A B C - 13482 A B C + 34608 A B C + 4692 A B C 5 7 4 8 4 7 4 6 2 4 5 3 + 3147 A C + 1680 A B + 5301 A B C + 20868 A B C - 13482 A B C 4 4 4 4 3 5 4 2 6 4 7 + 164808 A B C - 13482 A B C + 20868 A B C + 5301 A B C 4 8 3 9 3 8 3 7 2 3 6 3 + 1680 A C + 564 A B + 3780 A B C + 6510 A B C + 40080 A B C 3 5 4 3 4 5 3 3 6 3 2 7 - 13482 A B C - 13482 A B C + 40080 A B C + 6510 A B C 3 8 3 9 2 10 2 9 2 8 2 + 3780 A B C + 564 A C + 108 A B + 1362 A B C + 4200 A B C 2 7 3 2 6 4 2 5 5 2 4 6 + 6510 A B C + 20868 A B C + 34608 A B C + 20868 A B C 2 3 7 2 2 8 2 9 2 10 11 + 6510 A B C + 4200 A B C + 1362 A B C + 108 A C + 9 A B 10 9 2 8 3 7 4 6 5 + 216 A B C + 1362 A B C + 3780 A B C + 5301 A B C + 4692 A B C 5 6 4 7 3 8 2 9 10 + 4692 A B C + 5301 A B C + 3780 A B C + 1362 A B C + 216 A B C 11 11 10 2 9 3 8 4 7 5 + 9 A C + 9 B C + 108 B C + 564 B C + 1680 B C + 3147 B C 6 6 5 7 4 8 3 9 2 10 11 + 3864 B C + 3147 B C + 1680 B C + 564 B C + 108 B C + 9 B C 9 9 8 2 8 8 2 7 3 - 120 A B - 120 A C - 1200 A B - 2400 A B C - 1200 A C - 4980 A B 7 2 7 2 7 3 6 4 6 3 - 11580 A B C - 11580 A B C - 4980 A C - 11040 A B - 21060 A B C 6 2 2 6 3 6 4 5 5 - 20040 A B C - 21060 A B C - 11040 A C - 14280 A B 5 4 5 3 2 5 2 3 5 4 - 21000 A B C - 49140 A B C - 49140 A B C - 21000 A B C 5 5 4 6 4 5 4 4 2 - 14280 A C - 11040 A B - 21000 A B C - 78960 A B C 4 3 3 4 2 4 4 5 4 6 - 55020 A B C - 78960 A B C - 21000 A B C - 11040 A C 3 7 3 6 3 5 2 3 4 3 - 4980 A B - 21060 A B C - 49140 A B C - 55020 A B C 3 3 4 3 2 5 3 6 3 7 - 55020 A B C - 49140 A B C - 21060 A B C - 4980 A C 2 8 2 7 2 6 2 2 5 3 - 1200 A B - 11580 A B C - 20040 A B C - 49140 A B C 2 4 4 2 3 5 2 2 6 2 7 - 78960 A B C - 49140 A B C - 20040 A B C - 11580 A B C 2 8 9 8 7 2 6 3 - 1200 A C - 120 A B - 2400 A B C - 11580 A B C - 21060 A B C 5 4 4 5 3 6 2 7 - 21000 A B C - 21000 A B C - 21060 A B C - 11580 A B C 8 9 9 8 2 7 3 - 2400 A B C - 120 A C - 120 B C - 1200 B C - 4980 B C 6 4 5 5 4 6 3 7 2 8 - 11040 B C - 14280 B C - 11040 B C - 4980 B C - 1200 B C 9 7 7 6 2 6 6 2 - 120 B C + 399 A B + 399 A C + 3192 A B + 6384 A B C + 3192 A C 5 3 5 2 5 2 5 3 4 4 + 9177 A B + 19866 A B C + 19866 A B C + 9177 A C + 12768 A B 4 3 4 2 2 4 3 4 4 + 30471 A B C + 36624 A B C + 30471 A B C + 12768 A C 3 5 3 4 3 3 2 3 2 3 + 9177 A B + 30471 A B C + 39900 A B C + 39900 A B C 3 4 3 5 2 6 2 5 2 4 2 + 30471 A B C + 9177 A C + 3192 A B + 19866 A B C + 36624 A B C 2 3 3 2 2 4 2 5 2 6 7 + 39900 A B C + 36624 A B C + 19866 A B C + 3192 A C + 399 A B 6 5 2 4 3 3 4 + 6384 A B C + 19866 A B C + 30471 A B C + 30471 A B C 2 5 6 7 7 6 2 + 19866 A B C + 6384 A B C + 399 A C + 399 B C + 3192 B C 5 3 4 4 3 5 2 6 7 5 + 9177 B C + 12768 B C + 9177 B C + 3192 B C + 399 B C - 120 A B 5 4 2 4 4 2 3 3 - 120 A C - 480 A B - 2040 A B C - 480 A C - 720 A B 3 2 3 2 3 3 2 4 2 3 - 5520 A B C - 5520 A B C - 720 A C - 480 A B - 5520 A B C 2 2 2 2 3 2 4 5 4 - 10080 A B C - 5520 A B C - 480 A C - 120 A B - 2040 A B C 3 2 2 3 4 5 5 - 5520 A B C - 5520 A B C - 2040 A B C - 120 A C - 120 B C 4 2 3 3 2 4 5 4 3 3 - 480 B C - 720 B C - 480 B C - 120 B C + 16 A + 64 A B + 64 A C 2 2 2 2 2 3 2 2 + 96 A B + 192 A B C + 96 A C + 64 A B + 192 A B C + 192 A B C 3 4 3 2 2 3 4 / + 64 A C + 16 B + 64 B C + 96 B C + 64 B C + 16 C ) / (1120 / 5 (A + B + C) ) and in Maple notation 3/1120*A*B*C*(9*A^11*B+9*A^11*C+108*A^10*B^2+216*A^10*B*C+108*A^10*C^2+564*A^9* B^3+1362*A^9*B^2*C+1362*A^9*B*C^2+564*A^9*C^3+1680*A^8*B^4+3780*A^8*B^3*C+4200* A^8*B^2*C^2+3780*A^8*B*C^3+1680*A^8*C^4+3147*A^7*B^5+5301*A^7*B^4*C+6510*A^7*B^ 3*C^2+6510*A^7*B^2*C^3+5301*A^7*B*C^4+3147*A^7*C^5+3864*A^6*B^6+4692*A^6*B^5*C+ 20868*A^6*B^4*C^2+40080*A^6*B^3*C^3+20868*A^6*B^2*C^4+4692*A^6*B*C^5+3864*A^6*C ^6+3147*A^5*B^7+4692*A^5*B^6*C+34608*A^5*B^5*C^2-13482*A^5*B^4*C^3-13482*A^5*B^ 3*C^4+34608*A^5*B^2*C^5+4692*A^5*B*C^6+3147*A^5*C^7+1680*A^4*B^8+5301*A^4*B^7*C +20868*A^4*B^6*C^2-13482*A^4*B^5*C^3+164808*A^4*B^4*C^4-13482*A^4*B^3*C^5+20868 *A^4*B^2*C^6+5301*A^4*B*C^7+1680*A^4*C^8+564*A^3*B^9+3780*A^3*B^8*C+6510*A^3*B^ 7*C^2+40080*A^3*B^6*C^3-13482*A^3*B^5*C^4-13482*A^3*B^4*C^5+40080*A^3*B^3*C^6+ 6510*A^3*B^2*C^7+3780*A^3*B*C^8+564*A^3*C^9+108*A^2*B^10+1362*A^2*B^9*C+4200*A^ 2*B^8*C^2+6510*A^2*B^7*C^3+20868*A^2*B^6*C^4+34608*A^2*B^5*C^5+20868*A^2*B^4*C^ 6+6510*A^2*B^3*C^7+4200*A^2*B^2*C^8+1362*A^2*B*C^9+108*A^2*C^10+9*A*B^11+216*A* B^10*C+1362*A*B^9*C^2+3780*A*B^8*C^3+5301*A*B^7*C^4+4692*A*B^6*C^5+4692*A*B^5*C ^6+5301*A*B^4*C^7+3780*A*B^3*C^8+1362*A*B^2*C^9+216*A*B*C^10+9*A*C^11+9*B^11*C+ 108*B^10*C^2+564*B^9*C^3+1680*B^8*C^4+3147*B^7*C^5+3864*B^6*C^6+3147*B^5*C^7+ 1680*B^4*C^8+564*B^3*C^9+108*B^2*C^10+9*B*C^11-120*A^9*B-120*A^9*C-1200*A^8*B^2 -2400*A^8*B*C-1200*A^8*C^2-4980*A^7*B^3-11580*A^7*B^2*C-11580*A^7*B*C^2-4980*A^ 7*C^3-11040*A^6*B^4-21060*A^6*B^3*C-20040*A^6*B^2*C^2-21060*A^6*B*C^3-11040*A^6 *C^4-14280*A^5*B^5-21000*A^5*B^4*C-49140*A^5*B^3*C^2-49140*A^5*B^2*C^3-21000*A^ 5*B*C^4-14280*A^5*C^5-11040*A^4*B^6-21000*A^4*B^5*C-78960*A^4*B^4*C^2-55020*A^4 *B^3*C^3-78960*A^4*B^2*C^4-21000*A^4*B*C^5-11040*A^4*C^6-4980*A^3*B^7-21060*A^3 *B^6*C-49140*A^3*B^5*C^2-55020*A^3*B^4*C^3-55020*A^3*B^3*C^4-49140*A^3*B^2*C^5-\ 21060*A^3*B*C^6-4980*A^3*C^7-1200*A^2*B^8-11580*A^2*B^7*C-20040*A^2*B^6*C^2-\ 49140*A^2*B^5*C^3-78960*A^2*B^4*C^4-49140*A^2*B^3*C^5-20040*A^2*B^2*C^6-11580*A ^2*B*C^7-1200*A^2*C^8-120*A*B^9-2400*A*B^8*C-11580*A*B^7*C^2-21060*A*B^6*C^3-\ 21000*A*B^5*C^4-21000*A*B^4*C^5-21060*A*B^3*C^6-11580*A*B^2*C^7-2400*A*B*C^8-\ 120*A*C^9-120*B^9*C-1200*B^8*C^2-4980*B^7*C^3-11040*B^6*C^4-14280*B^5*C^5-11040 *B^4*C^6-4980*B^3*C^7-1200*B^2*C^8-120*B*C^9+399*A^7*B+399*A^7*C+3192*A^6*B^2+ 6384*A^6*B*C+3192*A^6*C^2+9177*A^5*B^3+19866*A^5*B^2*C+19866*A^5*B*C^2+9177*A^5 *C^3+12768*A^4*B^4+30471*A^4*B^3*C+36624*A^4*B^2*C^2+30471*A^4*B*C^3+12768*A^4* C^4+9177*A^3*B^5+30471*A^3*B^4*C+39900*A^3*B^3*C^2+39900*A^3*B^2*C^3+30471*A^3* B*C^4+9177*A^3*C^5+3192*A^2*B^6+19866*A^2*B^5*C+36624*A^2*B^4*C^2+39900*A^2*B^3 *C^3+36624*A^2*B^2*C^4+19866*A^2*B*C^5+3192*A^2*C^6+399*A*B^7+6384*A*B^6*C+ 19866*A*B^5*C^2+30471*A*B^4*C^3+30471*A*B^3*C^4+19866*A*B^2*C^5+6384*A*B*C^6+ 399*A*C^7+399*B^7*C+3192*B^6*C^2+9177*B^5*C^3+12768*B^4*C^4+9177*B^3*C^5+3192*B ^2*C^6+399*B*C^7-120*A^5*B-120*A^5*C-480*A^4*B^2-2040*A^4*B*C-480*A^4*C^2-720*A ^3*B^3-5520*A^3*B^2*C-5520*A^3*B*C^2-720*A^3*C^3-480*A^2*B^4-5520*A^2*B^3*C-\ 10080*A^2*B^2*C^2-5520*A^2*B*C^3-480*A^2*C^4-120*A*B^5-2040*A*B^4*C-5520*A*B^3* C^2-5520*A*B^2*C^3-2040*A*B*C^4-120*A*C^5-120*B^5*C-480*B^4*C^2-720*B^3*C^3-480 *B^2*C^4-120*B*C^5+16*A^4+64*A^3*B+64*A^3*C+96*A^2*B^2+192*A^2*B*C+96*A^2*C^2+ 64*A*B^3+192*A*B^2*C+192*A*B*C^2+64*A*C^3+16*B^4+64*B^3*C+96*B^2*C^2+64*B*C^3+ 16*C^4)/(A+B+C)^5 Hence the scaled, 5, -th moment about the mean is 11 11 10 2 10 10 2 9 3 A B C (9 A B + 9 A C + 108 A B + 216 A B C + 108 A C + 564 A B 9 2 9 2 9 3 8 4 8 3 + 1362 A B C + 1362 A B C + 564 A C + 1680 A B + 3780 A B C 8 2 2 8 3 8 4 7 5 7 4 + 4200 A B C + 3780 A B C + 1680 A C + 3147 A B + 5301 A B C 7 3 2 7 2 3 7 4 7 5 6 6 + 6510 A B C + 6510 A B C + 5301 A B C + 3147 A C + 3864 A B 6 5 6 4 2 6 3 3 6 2 4 + 4692 A B C + 20868 A B C + 40080 A B C + 20868 A B C 6 5 6 6 5 7 5 6 5 5 2 + 4692 A B C + 3864 A C + 3147 A B + 4692 A B C + 34608 A B C 5 4 3 5 3 4 5 2 5 5 6 - 13482 A B C - 13482 A B C + 34608 A B C + 4692 A B C 5 7 4 8 4 7 4 6 2 4 5 3 + 3147 A C + 1680 A B + 5301 A B C + 20868 A B C - 13482 A B C 4 4 4 4 3 5 4 2 6 4 7 + 164808 A B C - 13482 A B C + 20868 A B C + 5301 A B C 4 8 3 9 3 8 3 7 2 3 6 3 + 1680 A C + 564 A B + 3780 A B C + 6510 A B C + 40080 A B C 3 5 4 3 4 5 3 3 6 3 2 7 - 13482 A B C - 13482 A B C + 40080 A B C + 6510 A B C 3 8 3 9 2 10 2 9 2 8 2 + 3780 A B C + 564 A C + 108 A B + 1362 A B C + 4200 A B C 2 7 3 2 6 4 2 5 5 2 4 6 + 6510 A B C + 20868 A B C + 34608 A B C + 20868 A B C 2 3 7 2 2 8 2 9 2 10 11 + 6510 A B C + 4200 A B C + 1362 A B C + 108 A C + 9 A B 10 9 2 8 3 7 4 6 5 + 216 A B C + 1362 A B C + 3780 A B C + 5301 A B C + 4692 A B C 5 6 4 7 3 8 2 9 10 + 4692 A B C + 5301 A B C + 3780 A B C + 1362 A B C + 216 A B C 11 11 10 2 9 3 8 4 7 5 + 9 A C + 9 B C + 108 B C + 564 B C + 1680 B C + 3147 B C 6 6 5 7 4 8 3 9 2 10 11 + 3864 B C + 3147 B C + 1680 B C + 564 B C + 108 B C + 9 B C 9 9 8 2 8 8 2 7 3 - 120 A B - 120 A C - 1200 A B - 2400 A B C - 1200 A C - 4980 A B 7 2 7 2 7 3 6 4 6 3 - 11580 A B C - 11580 A B C - 4980 A C - 11040 A B - 21060 A B C 6 2 2 6 3 6 4 5 5 - 20040 A B C - 21060 A B C - 11040 A C - 14280 A B 5 4 5 3 2 5 2 3 5 4 - 21000 A B C - 49140 A B C - 49140 A B C - 21000 A B C 5 5 4 6 4 5 4 4 2 - 14280 A C - 11040 A B - 21000 A B C - 78960 A B C 4 3 3 4 2 4 4 5 4 6 - 55020 A B C - 78960 A B C - 21000 A B C - 11040 A C 3 7 3 6 3 5 2 3 4 3 - 4980 A B - 21060 A B C - 49140 A B C - 55020 A B C 3 3 4 3 2 5 3 6 3 7 - 55020 A B C - 49140 A B C - 21060 A B C - 4980 A C 2 8 2 7 2 6 2 2 5 3 - 1200 A B - 11580 A B C - 20040 A B C - 49140 A B C 2 4 4 2 3 5 2 2 6 2 7 - 78960 A B C - 49140 A B C - 20040 A B C - 11580 A B C 2 8 9 8 7 2 6 3 - 1200 A C - 120 A B - 2400 A B C - 11580 A B C - 21060 A B C 5 4 4 5 3 6 2 7 - 21000 A B C - 21000 A B C - 21060 A B C - 11580 A B C 8 9 9 8 2 7 3 - 2400 A B C - 120 A C - 120 B C - 1200 B C - 4980 B C 6 4 5 5 4 6 3 7 2 8 - 11040 B C - 14280 B C - 11040 B C - 4980 B C - 1200 B C 9 7 7 6 2 6 6 2 - 120 B C + 399 A B + 399 A C + 3192 A B + 6384 A B C + 3192 A C 5 3 5 2 5 2 5 3 4 4 + 9177 A B + 19866 A B C + 19866 A B C + 9177 A C + 12768 A B 4 3 4 2 2 4 3 4 4 + 30471 A B C + 36624 A B C + 30471 A B C + 12768 A C 3 5 3 4 3 3 2 3 2 3 + 9177 A B + 30471 A B C + 39900 A B C + 39900 A B C 3 4 3 5 2 6 2 5 2 4 2 + 30471 A B C + 9177 A C + 3192 A B + 19866 A B C + 36624 A B C 2 3 3 2 2 4 2 5 2 6 7 + 39900 A B C + 36624 A B C + 19866 A B C + 3192 A C + 399 A B 6 5 2 4 3 3 4 + 6384 A B C + 19866 A B C + 30471 A B C + 30471 A B C 2 5 6 7 7 6 2 + 19866 A B C + 6384 A B C + 399 A C + 399 B C + 3192 B C 5 3 4 4 3 5 2 6 7 5 + 9177 B C + 12768 B C + 9177 B C + 3192 B C + 399 B C - 120 A B 5 4 2 4 4 2 3 3 - 120 A C - 480 A B - 2040 A B C - 480 A C - 720 A B 3 2 3 2 3 3 2 4 2 3 - 5520 A B C - 5520 A B C - 720 A C - 480 A B - 5520 A B C 2 2 2 2 3 2 4 5 4 - 10080 A B C - 5520 A B C - 480 A C - 120 A B - 2040 A B C 3 2 2 3 4 5 5 - 5520 A B C - 5520 A B C - 2040 A B C - 120 A C - 120 B C 4 2 3 3 2 4 5 4 3 3 - 480 B C - 720 B C - 480 B C - 120 B C + 16 A + 64 A B + 64 A C 2 2 2 2 2 3 2 2 + 96 A B + 192 A B C + 96 A C + 64 A B + 192 A B C + 192 A B C / 3 4 3 2 2 3 4 1/2 1/2 / | + 64 A C + 16 B + 64 B C + 96 B C + 64 B C + 16 C ) 3 2 / | / | \ 5 2520 (A + B + C) / 2 2 2 2 2 2 \5/2\ |A B C (A B + A C + A B - 3 A B C + A C + B C + B C - A - B - C)| | |---------------------------------------------------------------------| | | 2 | | \ (A + B + C) / / and in Maple notation 1/2520*A*B*C*(9*A^11*B+9*A^11*C+108*A^10*B^2+216*A^10*B*C+108*A^10*C^2+564*A^9* B^3+1362*A^9*B^2*C+1362*A^9*B*C^2+564*A^9*C^3+1680*A^8*B^4+3780*A^8*B^3*C+4200* A^8*B^2*C^2+3780*A^8*B*C^3+1680*A^8*C^4+3147*A^7*B^5+5301*A^7*B^4*C+6510*A^7*B^ 3*C^2+6510*A^7*B^2*C^3+5301*A^7*B*C^4+3147*A^7*C^5+3864*A^6*B^6+4692*A^6*B^5*C+ 20868*A^6*B^4*C^2+40080*A^6*B^3*C^3+20868*A^6*B^2*C^4+4692*A^6*B*C^5+3864*A^6*C ^6+3147*A^5*B^7+4692*A^5*B^6*C+34608*A^5*B^5*C^2-13482*A^5*B^4*C^3-13482*A^5*B^ 3*C^4+34608*A^5*B^2*C^5+4692*A^5*B*C^6+3147*A^5*C^7+1680*A^4*B^8+5301*A^4*B^7*C +20868*A^4*B^6*C^2-13482*A^4*B^5*C^3+164808*A^4*B^4*C^4-13482*A^4*B^3*C^5+20868 *A^4*B^2*C^6+5301*A^4*B*C^7+1680*A^4*C^8+564*A^3*B^9+3780*A^3*B^8*C+6510*A^3*B^ 7*C^2+40080*A^3*B^6*C^3-13482*A^3*B^5*C^4-13482*A^3*B^4*C^5+40080*A^3*B^3*C^6+ 6510*A^3*B^2*C^7+3780*A^3*B*C^8+564*A^3*C^9+108*A^2*B^10+1362*A^2*B^9*C+4200*A^ 2*B^8*C^2+6510*A^2*B^7*C^3+20868*A^2*B^6*C^4+34608*A^2*B^5*C^5+20868*A^2*B^4*C^ 6+6510*A^2*B^3*C^7+4200*A^2*B^2*C^8+1362*A^2*B*C^9+108*A^2*C^10+9*A*B^11+216*A* B^10*C+1362*A*B^9*C^2+3780*A*B^8*C^3+5301*A*B^7*C^4+4692*A*B^6*C^5+4692*A*B^5*C ^6+5301*A*B^4*C^7+3780*A*B^3*C^8+1362*A*B^2*C^9+216*A*B*C^10+9*A*C^11+9*B^11*C+ 108*B^10*C^2+564*B^9*C^3+1680*B^8*C^4+3147*B^7*C^5+3864*B^6*C^6+3147*B^5*C^7+ 1680*B^4*C^8+564*B^3*C^9+108*B^2*C^10+9*B*C^11-120*A^9*B-120*A^9*C-1200*A^8*B^2 -2400*A^8*B*C-1200*A^8*C^2-4980*A^7*B^3-11580*A^7*B^2*C-11580*A^7*B*C^2-4980*A^ 7*C^3-11040*A^6*B^4-21060*A^6*B^3*C-20040*A^6*B^2*C^2-21060*A^6*B*C^3-11040*A^6 *C^4-14280*A^5*B^5-21000*A^5*B^4*C-49140*A^5*B^3*C^2-49140*A^5*B^2*C^3-21000*A^ 5*B*C^4-14280*A^5*C^5-11040*A^4*B^6-21000*A^4*B^5*C-78960*A^4*B^4*C^2-55020*A^4 *B^3*C^3-78960*A^4*B^2*C^4-21000*A^4*B*C^5-11040*A^4*C^6-4980*A^3*B^7-21060*A^3 *B^6*C-49140*A^3*B^5*C^2-55020*A^3*B^4*C^3-55020*A^3*B^3*C^4-49140*A^3*B^2*C^5-\ 21060*A^3*B*C^6-4980*A^3*C^7-1200*A^2*B^8-11580*A^2*B^7*C-20040*A^2*B^6*C^2-\ 49140*A^2*B^5*C^3-78960*A^2*B^4*C^4-49140*A^2*B^3*C^5-20040*A^2*B^2*C^6-11580*A ^2*B*C^7-1200*A^2*C^8-120*A*B^9-2400*A*B^8*C-11580*A*B^7*C^2-21060*A*B^6*C^3-\ 21000*A*B^5*C^4-21000*A*B^4*C^5-21060*A*B^3*C^6-11580*A*B^2*C^7-2400*A*B*C^8-\ 120*A*C^9-120*B^9*C-1200*B^8*C^2-4980*B^7*C^3-11040*B^6*C^4-14280*B^5*C^5-11040 *B^4*C^6-4980*B^3*C^7-1200*B^2*C^8-120*B*C^9+399*A^7*B+399*A^7*C+3192*A^6*B^2+ 6384*A^6*B*C+3192*A^6*C^2+9177*A^5*B^3+19866*A^5*B^2*C+19866*A^5*B*C^2+9177*A^5 *C^3+12768*A^4*B^4+30471*A^4*B^3*C+36624*A^4*B^2*C^2+30471*A^4*B*C^3+12768*A^4* C^4+9177*A^3*B^5+30471*A^3*B^4*C+39900*A^3*B^3*C^2+39900*A^3*B^2*C^3+30471*A^3* B*C^4+9177*A^3*C^5+3192*A^2*B^6+19866*A^2*B^5*C+36624*A^2*B^4*C^2+39900*A^2*B^3 *C^3+36624*A^2*B^2*C^4+19866*A^2*B*C^5+3192*A^2*C^6+399*A*B^7+6384*A*B^6*C+ 19866*A*B^5*C^2+30471*A*B^4*C^3+30471*A*B^3*C^4+19866*A*B^2*C^5+6384*A*B*C^6+ 399*A*C^7+399*B^7*C+3192*B^6*C^2+9177*B^5*C^3+12768*B^4*C^4+9177*B^3*C^5+3192*B ^2*C^6+399*B*C^7-120*A^5*B-120*A^5*C-480*A^4*B^2-2040*A^4*B*C-480*A^4*C^2-720*A ^3*B^3-5520*A^3*B^2*C-5520*A^3*B*C^2-720*A^3*C^3-480*A^2*B^4-5520*A^2*B^3*C-\ 10080*A^2*B^2*C^2-5520*A^2*B*C^3-480*A^2*C^4-120*A*B^5-2040*A*B^4*C-5520*A*B^3* C^2-5520*A*B^2*C^3-2040*A*B*C^4-120*A*C^5-120*B^5*C-480*B^4*C^2-720*B^3*C^3-480 *B^2*C^4-120*B*C^5+16*A^4+64*A^3*B+64*A^3*C+96*A^2*B^2+192*A^2*B*C+96*A^2*C^2+ 64*A*B^3+192*A*B^2*C+192*A*B*C^2+64*A*C^3+16*B^4+64*B^3*C+96*B^2*C^2+64*B*C^3+ 16*C^4)/(A+B+C)^5*3^(1/2)*2^(1/2)/(A*B*C*(A^2*B+A^2*C+A*B^2-3*A*B*C+A*C^2+B^2*C +B*C^2-A-B-C)/(A+B+C)^2)^(5/2) Note that if N is large and the starting capitals are x1*N,x2*N,x3*N, (where\ , of course, x1+x2+x3=1), then the limit of the, 5, -th scaled moment about the mean tends to 12 2 1/2 1/2 11 3 1/2 1/2 10 4 1/2 1/2 (3 x1 x2 x3 3 2 + 36 x1 x2 x3 3 2 + 188 x1 x2 x3 3 2 9 5 1/2 1/2 8 6 1/2 1/2 + 560 x1 x2 x3 3 2 + 1049 x1 x2 x3 3 2 7 7 1/2 1/2 6 8 1/2 1/2 + 1288 x1 x2 x3 3 2 + 1049 x1 x2 x3 3 2 5 9 1/2 1/2 4 10 1/2 1/2 + 560 x1 x2 x3 3 2 + 188 x1 x2 x3 3 2 3 11 1/2 1/2 2 12 1/2 1/2 + 36 x1 x2 x3 3 2 + 3 x1 x2 x3 3 2 4 2 9 1/2 1/2 4 10 1/2 1/2 + 1260 x1 x2 x3 3 2 + 188 x1 x2 x3 3 2 3 11 1/2 1/2 3 10 2 1/2 1/2 + 36 x1 x2 x3 3 2 + 454 x1 x2 x3 3 2 3 9 3 1/2 1/2 3 8 4 1/2 1/2 + 1400 x1 x2 x3 3 2 + 2170 x1 x2 x3 3 2 3 7 5 1/2 1/2 3 6 6 1/2 1/2 + 6956 x1 x2 x3 3 2 + 11536 x1 x2 x3 3 2 3 5 7 1/2 1/2 3 4 8 1/2 1/2 + 6956 x1 x2 x3 3 2 + 2170 x1 x2 x3 3 2 3 3 9 1/2 1/2 3 2 10 1/2 1/2 + 1400 x1 x2 x3 3 2 + 454 x1 x2 x3 3 2 3 11 1/2 1/2 2 12 1/2 1/2 + 36 x1 x2 x3 3 2 + 3 x1 x2 x3 3 2 2 11 2 1/2 1/2 2 10 3 1/2 1/2 + 72 x1 x2 x3 3 2 + 454 x1 x2 x3 3 2 2 9 4 1/2 1/2 2 8 5 1/2 1/2 + 1260 x1 x2 x3 3 2 + 1767 x1 x2 x3 3 2 2 7 6 1/2 1/2 2 6 7 1/2 1/2 + 1564 x1 x2 x3 3 2 + 1564 x1 x2 x3 3 2 2 5 8 1/2 1/2 2 4 9 1/2 1/2 + 1767 x1 x2 x3 3 2 + 1260 x1 x2 x3 3 2 2 3 10 1/2 1/2 2 2 11 1/2 1/2 + 454 x1 x2 x3 3 2 + 72 x1 x2 x3 3 2 2 12 1/2 1/2 12 2 1/2 1/2 + 3 x1 x2 x3 3 2 + 3 x1 x2 x3 3 2 12 2 1/2 1/2 11 3 1/2 1/2 + 3 x1 x2 x3 3 2 + 36 x1 x2 x3 3 2 11 2 2 1/2 1/2 11 3 1/2 1/2 + 72 x1 x2 x3 3 2 + 36 x1 x2 x3 3 2 10 4 1/2 1/2 10 3 2 1/2 1/2 + 188 x1 x2 x3 3 2 + 454 x1 x2 x3 3 2 10 2 3 1/2 1/2 10 4 1/2 1/2 + 454 x1 x2 x3 3 2 + 188 x1 x2 x3 3 2 9 5 1/2 1/2 9 4 2 1/2 1/2 + 560 x1 x2 x3 3 2 + 1260 x1 x2 x3 3 2 9 3 3 1/2 1/2 9 2 4 1/2 1/2 + 1400 x1 x2 x3 3 2 + 1260 x1 x2 x3 3 2 9 5 1/2 1/2 8 6 1/2 1/2 + 560 x1 x2 x3 3 2 + 1049 x1 x2 x3 3 2 8 5 2 1/2 1/2 8 4 3 1/2 1/2 + 1767 x1 x2 x3 3 2 + 2170 x1 x2 x3 3 2 8 3 4 1/2 1/2 8 2 5 1/2 1/2 + 2170 x1 x2 x3 3 2 + 1767 x1 x2 x3 3 2 8 6 1/2 1/2 7 7 1/2 1/2 + 1049 x1 x2 x3 3 2 + 1288 x1 x2 x3 3 2 7 6 2 1/2 1/2 7 5 3 1/2 1/2 + 1564 x1 x2 x3 3 2 + 6956 x1 x2 x3 3 2 7 4 4 1/2 1/2 7 3 5 1/2 1/2 + 13360 x1 x2 x3 3 2 + 6956 x1 x2 x3 3 2 7 2 6 1/2 1/2 7 7 1/2 1/2 + 1564 x1 x2 x3 3 2 + 1288 x1 x2 x3 3 2 6 8 1/2 1/2 6 7 2 1/2 1/2 + 1049 x1 x2 x3 3 2 + 1564 x1 x2 x3 3 2 6 6 3 1/2 1/2 6 5 4 1/2 1/2 + 11536 x1 x2 x3 3 2 - 4494 x1 x2 x3 3 2 6 4 5 1/2 1/2 6 3 6 1/2 1/2 - 4494 x1 x2 x3 3 2 + 11536 x1 x2 x3 3 2 6 2 7 1/2 1/2 6 8 1/2 1/2 + 1564 x1 x2 x3 3 2 + 1049 x1 x2 x3 3 2 5 9 1/2 1/2 5 8 2 1/2 1/2 + 560 x1 x2 x3 3 2 + 1767 x1 x2 x3 3 2 5 7 3 1/2 1/2 5 6 4 1/2 1/2 + 6956 x1 x2 x3 3 2 - 4494 x1 x2 x3 3 2 5 5 5 1/2 1/2 5 4 6 1/2 1/2 + 54936 x1 x2 x3 3 2 - 4494 x1 x2 x3 3 2 5 3 7 1/2 1/2 5 2 8 1/2 1/2 + 6956 x1 x2 x3 3 2 + 1767 x1 x2 x3 3 2 5 9 1/2 1/2 4 10 1/2 1/2 + 560 x1 x2 x3 3 2 + 188 x1 x2 x3 3 2 4 9 2 1/2 1/2 4 8 3 1/2 1/2 + 1260 x1 x2 x3 3 2 + 2170 x1 x2 x3 3 2 4 7 4 1/2 1/2 4 6 5 1/2 1/2 + 13360 x1 x2 x3 3 2 - 4494 x1 x2 x3 3 2 4 5 6 1/2 1/2 4 4 7 1/2 1/2 - 4494 x1 x2 x3 3 2 + 13360 x1 x2 x3 3 2 4 3 8 1/2 1/2 / 5 4 + 2170 x1 x2 x3 3 2 ) / (840 %1 x1 + 4200 %1 x1 x2 / 4 3 2 3 3 2 + 4200 %1 x1 x3 + 8400 %1 x1 x2 + 16800 %1 x1 x2 x3 + 8400 %1 x1 x3 2 3 2 2 2 2 + 8400 %1 x1 x2 + 25200 %1 x1 x2 x3 + 25200 %1 x1 x2 x3 2 3 4 3 + 8400 %1 x1 x3 + 4200 %1 x1 x2 + 16800 %1 x1 x2 x3 2 2 3 4 5 + 25200 %1 x1 x2 x3 + 16800 %1 x1 x2 x3 + 4200 %1 x1 x3 + 840 %1 x2 4 3 2 2 3 4 + 4200 %1 x2 x3 + 8400 %1 x2 x3 + 8400 %1 x2 x3 + 4200 %1 x2 x3 5 + 840 %1 x3 ) / %1 := |x1 x2 x3 \ 2 2 2 2 2 2 / (x1 x2 + x1 x3 + x1 x2 - 3 x1 x2 x3 + x1 x3 + x2 x3 + x2 x3 ) / ( / 2 2 2 \(5/2) x1 + 2 x1 x2 + 2 x1 x3 + x2 + 2 x2 x3 + x3 )| / and in Maple notation (3*x1*x2^12*x3^2*3^(1/2)*2^(1/2)+36*x1*x2^11*x3^3*3^(1/2)*2^(1/2)+188*x1*x2^10* x3^4*3^(1/2)*2^(1/2)+560*x1*x2^9*x3^5*3^(1/2)*2^(1/2)+1049*x1*x2^8*x3^6*3^(1/2) *2^(1/2)+1288*x1*x2^7*x3^7*3^(1/2)*2^(1/2)+1049*x1*x2^6*x3^8*3^(1/2)*2^(1/2)+ 560*x1*x2^5*x3^9*3^(1/2)*2^(1/2)+188*x1*x2^4*x3^10*3^(1/2)*2^(1/2)+36*x1*x2^3* x3^11*3^(1/2)*2^(1/2)+3*x1*x2^2*x3^12*3^(1/2)*2^(1/2)+1260*x1^4*x2^2*x3^9*3^(1/ 2)*2^(1/2)+188*x1^4*x2*x3^10*3^(1/2)*2^(1/2)+36*x1^3*x2^11*x3*3^(1/2)*2^(1/2)+ 454*x1^3*x2^10*x3^2*3^(1/2)*2^(1/2)+1400*x1^3*x2^9*x3^3*3^(1/2)*2^(1/2)+2170*x1 ^3*x2^8*x3^4*3^(1/2)*2^(1/2)+6956*x1^3*x2^7*x3^5*3^(1/2)*2^(1/2)+11536*x1^3*x2^ 6*x3^6*3^(1/2)*2^(1/2)+6956*x1^3*x2^5*x3^7*3^(1/2)*2^(1/2)+2170*x1^3*x2^4*x3^8* 3^(1/2)*2^(1/2)+1400*x1^3*x2^3*x3^9*3^(1/2)*2^(1/2)+454*x1^3*x2^2*x3^10*3^(1/2) *2^(1/2)+36*x1^3*x2*x3^11*3^(1/2)*2^(1/2)+3*x1^2*x2^12*x3*3^(1/2)*2^(1/2)+72*x1 ^2*x2^11*x3^2*3^(1/2)*2^(1/2)+454*x1^2*x2^10*x3^3*3^(1/2)*2^(1/2)+1260*x1^2*x2^ 9*x3^4*3^(1/2)*2^(1/2)+1767*x1^2*x2^8*x3^5*3^(1/2)*2^(1/2)+1564*x1^2*x2^7*x3^6* 3^(1/2)*2^(1/2)+1564*x1^2*x2^6*x3^7*3^(1/2)*2^(1/2)+1767*x1^2*x2^5*x3^8*3^(1/2) *2^(1/2)+1260*x1^2*x2^4*x3^9*3^(1/2)*2^(1/2)+454*x1^2*x2^3*x3^10*3^(1/2)*2^(1/2 )+72*x1^2*x2^2*x3^11*3^(1/2)*2^(1/2)+3*x1^2*x2*x3^12*3^(1/2)*2^(1/2)+3*x1^12*x2 ^2*x3*3^(1/2)*2^(1/2)+3*x1^12*x2*x3^2*3^(1/2)*2^(1/2)+36*x1^11*x2^3*x3*3^(1/2)* 2^(1/2)+72*x1^11*x2^2*x3^2*3^(1/2)*2^(1/2)+36*x1^11*x2*x3^3*3^(1/2)*2^(1/2)+188 *x1^10*x2^4*x3*3^(1/2)*2^(1/2)+454*x1^10*x2^3*x3^2*3^(1/2)*2^(1/2)+454*x1^10*x2 ^2*x3^3*3^(1/2)*2^(1/2)+188*x1^10*x2*x3^4*3^(1/2)*2^(1/2)+560*x1^9*x2^5*x3*3^(1 /2)*2^(1/2)+1260*x1^9*x2^4*x3^2*3^(1/2)*2^(1/2)+1400*x1^9*x2^3*x3^3*3^(1/2)*2^( 1/2)+1260*x1^9*x2^2*x3^4*3^(1/2)*2^(1/2)+560*x1^9*x2*x3^5*3^(1/2)*2^(1/2)+1049* x1^8*x2^6*x3*3^(1/2)*2^(1/2)+1767*x1^8*x2^5*x3^2*3^(1/2)*2^(1/2)+2170*x1^8*x2^4 *x3^3*3^(1/2)*2^(1/2)+2170*x1^8*x2^3*x3^4*3^(1/2)*2^(1/2)+1767*x1^8*x2^2*x3^5*3 ^(1/2)*2^(1/2)+1049*x1^8*x2*x3^6*3^(1/2)*2^(1/2)+1288*x1^7*x2^7*x3*3^(1/2)*2^(1 /2)+1564*x1^7*x2^6*x3^2*3^(1/2)*2^(1/2)+6956*x1^7*x2^5*x3^3*3^(1/2)*2^(1/2)+ 13360*x1^7*x2^4*x3^4*3^(1/2)*2^(1/2)+6956*x1^7*x2^3*x3^5*3^(1/2)*2^(1/2)+1564* x1^7*x2^2*x3^6*3^(1/2)*2^(1/2)+1288*x1^7*x2*x3^7*3^(1/2)*2^(1/2)+1049*x1^6*x2^8 *x3*3^(1/2)*2^(1/2)+1564*x1^6*x2^7*x3^2*3^(1/2)*2^(1/2)+11536*x1^6*x2^6*x3^3*3^ (1/2)*2^(1/2)-4494*x1^6*x2^5*x3^4*3^(1/2)*2^(1/2)-4494*x1^6*x2^4*x3^5*3^(1/2)*2 ^(1/2)+11536*x1^6*x2^3*x3^6*3^(1/2)*2^(1/2)+1564*x1^6*x2^2*x3^7*3^(1/2)*2^(1/2) +1049*x1^6*x2*x3^8*3^(1/2)*2^(1/2)+560*x1^5*x2^9*x3*3^(1/2)*2^(1/2)+1767*x1^5* x2^8*x3^2*3^(1/2)*2^(1/2)+6956*x1^5*x2^7*x3^3*3^(1/2)*2^(1/2)-4494*x1^5*x2^6*x3 ^4*3^(1/2)*2^(1/2)+54936*x1^5*x2^5*x3^5*3^(1/2)*2^(1/2)-4494*x1^5*x2^4*x3^6*3^( 1/2)*2^(1/2)+6956*x1^5*x2^3*x3^7*3^(1/2)*2^(1/2)+1767*x1^5*x2^2*x3^8*3^(1/2)*2^ (1/2)+560*x1^5*x2*x3^9*3^(1/2)*2^(1/2)+188*x1^4*x2^10*x3*3^(1/2)*2^(1/2)+1260* x1^4*x2^9*x3^2*3^(1/2)*2^(1/2)+2170*x1^4*x2^8*x3^3*3^(1/2)*2^(1/2)+13360*x1^4* x2^7*x3^4*3^(1/2)*2^(1/2)-4494*x1^4*x2^6*x3^5*3^(1/2)*2^(1/2)-4494*x1^4*x2^5*x3 ^6*3^(1/2)*2^(1/2)+13360*x1^4*x2^4*x3^7*3^(1/2)*2^(1/2)+2170*x1^4*x2^3*x3^8*3^( 1/2)*2^(1/2))/(840*(x1*x2*x3*(x1^2*x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2* x3+x2*x3^2)/(x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2*x3+x3^2))^(5/2)*x1^5+4200*(x1*x2*x3 *(x1^2*x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2*x3+x2*x3^2)/(x1^2+2*x1*x2+2* x1*x3+x2^2+2*x2*x3+x3^2))^(5/2)*x1^4*x2+4200*(x1*x2*x3*(x1^2*x2+x1^2*x3+x1*x2^2 -3*x1*x2*x3+x1*x3^2+x2^2*x3+x2*x3^2)/(x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2*x3+x3^2))^ (5/2)*x1^4*x3+8400*(x1*x2*x3*(x1^2*x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2* x3+x2*x3^2)/(x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2*x3+x3^2))^(5/2)*x1^3*x2^2+16800*(x1 *x2*x3*(x1^2*x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2*x3+x2*x3^2)/(x1^2+2*x1* x2+2*x1*x3+x2^2+2*x2*x3+x3^2))^(5/2)*x1^3*x2*x3+8400*(x1*x2*x3*(x1^2*x2+x1^2*x3 +x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2*x3+x2*x3^2)/(x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2*x3 +x3^2))^(5/2)*x1^3*x3^2+8400*(x1*x2*x3*(x1^2*x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+x1* x3^2+x2^2*x3+x2*x3^2)/(x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2*x3+x3^2))^(5/2)*x1^2*x2^3 +25200*(x1*x2*x3*(x1^2*x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2*x3+x2*x3^2)/( x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2*x3+x3^2))^(5/2)*x1^2*x2^2*x3+25200*(x1*x2*x3*(x1 ^2*x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2*x3+x2*x3^2)/(x1^2+2*x1*x2+2*x1*x3 +x2^2+2*x2*x3+x3^2))^(5/2)*x1^2*x2*x3^2+8400*(x1*x2*x3*(x1^2*x2+x1^2*x3+x1*x2^2 -3*x1*x2*x3+x1*x3^2+x2^2*x3+x2*x3^2)/(x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2*x3+x3^2))^ (5/2)*x1^2*x3^3+4200*(x1*x2*x3*(x1^2*x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2 *x3+x2*x3^2)/(x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2*x3+x3^2))^(5/2)*x1*x2^4+16800*(x1* x2*x3*(x1^2*x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2*x3+x2*x3^2)/(x1^2+2*x1* x2+2*x1*x3+x2^2+2*x2*x3+x3^2))^(5/2)*x1*x2^3*x3+25200*(x1*x2*x3*(x1^2*x2+x1^2* x3+x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2*x3+x2*x3^2)/(x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2* x3+x3^2))^(5/2)*x1*x2^2*x3^2+16800*(x1*x2*x3*(x1^2*x2+x1^2*x3+x1*x2^2-3*x1*x2* x3+x1*x3^2+x2^2*x3+x2*x3^2)/(x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2*x3+x3^2))^(5/2)*x1* x2*x3^3+4200*(x1*x2*x3*(x1^2*x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2*x3+x2* x3^2)/(x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2*x3+x3^2))^(5/2)*x1*x3^4+840*(x1*x2*x3*(x1 ^2*x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2*x3+x2*x3^2)/(x1^2+2*x1*x2+2*x1*x3 +x2^2+2*x2*x3+x3^2))^(5/2)*x2^5+4200*(x1*x2*x3*(x1^2*x2+x1^2*x3+x1*x2^2-3*x1*x2 *x3+x1*x3^2+x2^2*x3+x2*x3^2)/(x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2*x3+x3^2))^(5/2)*x2 ^4*x3+8400*(x1*x2*x3*(x1^2*x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2*x3+x2*x3^ 2)/(x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2*x3+x3^2))^(5/2)*x2^3*x3^2+8400*(x1*x2*x3*(x1 ^2*x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2*x3+x2*x3^2)/(x1^2+2*x1*x2+2*x1*x3 +x2^2+2*x2*x3+x3^2))^(5/2)*x2^2*x3^3+4200*(x1*x2*x3*(x1^2*x2+x1^2*x3+x1*x2^2-3* x1*x2*x3+x1*x3^2+x2^2*x3+x2*x3^2)/(x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2*x3+x3^2))^(5/ 2)*x2*x3^4+840*(x1*x2*x3*(x1^2*x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2*x3+x2 *x3^2)/(x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2*x3+x3^2))^(5/2)*x3^5) The , 6, -th (straight) moment of the Duration is: 9 9 8 2 8 8 2 7 3 3 A B C (90 A B + 90 A C + 900 A B + 1800 A B C + 900 A C + 3753 A B 7 2 7 2 7 3 6 4 6 3 + 11556 A B C + 11556 A B C + 3753 A C + 8424 A B + 36072 A B C 6 2 2 6 3 6 4 5 5 + 55296 A B C + 36072 A B C + 8424 A C + 10980 A B 5 4 5 3 2 5 2 3 5 4 + 62424 A B C + 132768 A B C + 132768 A B C + 62424 A B C 5 5 4 6 4 5 4 4 2 + 10980 A C + 8424 A B + 62424 A B C + 176256 A B C 4 3 3 4 2 4 4 5 4 6 + 244512 A B C + 176256 A B C + 62424 A B C + 8424 A C 3 7 3 6 3 5 2 3 4 3 + 3753 A B + 36072 A B C + 132768 A B C + 244512 A B C 3 3 4 3 2 5 3 6 3 7 + 244512 A B C + 132768 A B C + 36072 A B C + 3753 A C 2 8 2 7 2 6 2 2 5 3 + 900 A B + 11556 A B C + 55296 A B C + 132768 A B C 2 4 4 2 3 5 2 2 6 2 7 + 176256 A B C + 132768 A B C + 55296 A B C + 11556 A B C 2 8 9 8 7 2 6 3 + 900 A C + 90 A B + 1800 A B C + 11556 A B C + 36072 A B C 5 4 4 5 3 6 2 7 + 62424 A B C + 62424 A B C + 36072 A B C + 11556 A B C 8 9 9 8 2 7 3 6 4 + 1800 A B C + 90 A C + 90 B C + 900 B C + 3753 B C + 8424 B C 5 5 4 6 3 7 2 8 9 7 + 10980 B C + 8424 B C + 3753 B C + 900 B C + 90 B C - 1485 A B 7 6 2 6 6 2 5 3 - 1485 A C - 11880 A B - 23760 A B C - 11880 A C - 36630 A B 5 2 5 2 5 3 4 4 - 114840 A B C - 114840 A B C - 36630 A C - 53460 A B 4 3 4 2 2 4 3 4 4 - 243540 A B C - 380160 A B C - 243540 A B C - 53460 A C 3 5 3 4 3 3 2 3 2 3 - 36630 A B - 243540 A B C - 554400 A B C - 554400 A B C 3 4 3 5 2 6 2 5 - 243540 A B C - 36630 A C - 11880 A B - 114840 A B C 2 4 2 2 3 3 2 2 4 2 5 - 380160 A B C - 554400 A B C - 380160 A B C - 114840 A B C 2 6 7 6 5 2 4 3 - 11880 A C - 1485 A B - 23760 A B C - 114840 A B C - 243540 A B C 3 4 2 5 6 7 7 - 243540 A B C - 114840 A B C - 23760 A B C - 1485 A C - 1485 B C 6 2 5 3 4 4 3 5 2 6 - 11880 B C - 36630 B C - 53460 B C - 36630 B C - 11880 B C 7 5 5 4 2 4 - 1485 B C + 6996 A B + 6996 A C + 41976 A B + 83952 A B C 4 2 3 3 3 2 3 2 + 41976 A C + 80454 A B + 265848 A B C + 265848 A B C 3 3 2 4 2 3 2 2 2 + 80454 A C + 41976 A B + 265848 A B C + 447744 A B C 2 3 2 4 5 4 3 2 + 265848 A B C + 41976 A C + 6996 A B + 83952 A B C + 265848 A B C 2 3 4 5 5 4 2 + 265848 A B C + 83952 A B C + 6996 A C + 6996 B C + 41976 B C 3 3 2 4 5 3 3 + 80454 B C + 41976 B C + 6996 B C - 7425 A B - 7425 A C 2 2 2 2 2 3 2 - 29700 A B - 59400 A B C - 29700 A C - 7425 A B - 59400 A B C 2 3 3 2 2 3 - 59400 A B C - 7425 A C - 7425 B C - 29700 B C - 7425 B C - 2992 A B - 2992 A C - 2992 B C - 1280)/(24640 (A + B + C)) and in Maple notation 3/24640*A*B*C*(90*A^9*B+90*A^9*C+900*A^8*B^2+1800*A^8*B*C+900*A^8*C^2+3753*A^7* B^3+11556*A^7*B^2*C+11556*A^7*B*C^2+3753*A^7*C^3+8424*A^6*B^4+36072*A^6*B^3*C+ 55296*A^6*B^2*C^2+36072*A^6*B*C^3+8424*A^6*C^4+10980*A^5*B^5+62424*A^5*B^4*C+ 132768*A^5*B^3*C^2+132768*A^5*B^2*C^3+62424*A^5*B*C^4+10980*A^5*C^5+8424*A^4*B^ 6+62424*A^4*B^5*C+176256*A^4*B^4*C^2+244512*A^4*B^3*C^3+176256*A^4*B^2*C^4+ 62424*A^4*B*C^5+8424*A^4*C^6+3753*A^3*B^7+36072*A^3*B^6*C+132768*A^3*B^5*C^2+ 244512*A^3*B^4*C^3+244512*A^3*B^3*C^4+132768*A^3*B^2*C^5+36072*A^3*B*C^6+3753*A ^3*C^7+900*A^2*B^8+11556*A^2*B^7*C+55296*A^2*B^6*C^2+132768*A^2*B^5*C^3+176256* A^2*B^4*C^4+132768*A^2*B^3*C^5+55296*A^2*B^2*C^6+11556*A^2*B*C^7+900*A^2*C^8+90 *A*B^9+1800*A*B^8*C+11556*A*B^7*C^2+36072*A*B^6*C^3+62424*A*B^5*C^4+62424*A*B^4 *C^5+36072*A*B^3*C^6+11556*A*B^2*C^7+1800*A*B*C^8+90*A*C^9+90*B^9*C+900*B^8*C^2 +3753*B^7*C^3+8424*B^6*C^4+10980*B^5*C^5+8424*B^4*C^6+3753*B^3*C^7+900*B^2*C^8+ 90*B*C^9-1485*A^7*B-1485*A^7*C-11880*A^6*B^2-23760*A^6*B*C-11880*A^6*C^2-36630* A^5*B^3-114840*A^5*B^2*C-114840*A^5*B*C^2-36630*A^5*C^3-53460*A^4*B^4-243540*A^ 4*B^3*C-380160*A^4*B^2*C^2-243540*A^4*B*C^3-53460*A^4*C^4-36630*A^3*B^5-243540* A^3*B^4*C-554400*A^3*B^3*C^2-554400*A^3*B^2*C^3-243540*A^3*B*C^4-36630*A^3*C^5-\ 11880*A^2*B^6-114840*A^2*B^5*C-380160*A^2*B^4*C^2-554400*A^2*B^3*C^3-380160*A^2 *B^2*C^4-114840*A^2*B*C^5-11880*A^2*C^6-1485*A*B^7-23760*A*B^6*C-114840*A*B^5*C ^2-243540*A*B^4*C^3-243540*A*B^3*C^4-114840*A*B^2*C^5-23760*A*B*C^6-1485*A*C^7-\ 1485*B^7*C-11880*B^6*C^2-36630*B^5*C^3-53460*B^4*C^4-36630*B^3*C^5-11880*B^2*C^ 6-1485*B*C^7+6996*A^5*B+6996*A^5*C+41976*A^4*B^2+83952*A^4*B*C+41976*A^4*C^2+ 80454*A^3*B^3+265848*A^3*B^2*C+265848*A^3*B*C^2+80454*A^3*C^3+41976*A^2*B^4+ 265848*A^2*B^3*C+447744*A^2*B^2*C^2+265848*A^2*B*C^3+41976*A^2*C^4+6996*A*B^5+ 83952*A*B^4*C+265848*A*B^3*C^2+265848*A*B^2*C^3+83952*A*B*C^4+6996*A*C^5+6996*B ^5*C+41976*B^4*C^2+80454*B^3*C^3+41976*B^2*C^4+6996*B*C^5-7425*A^3*B-7425*A^3*C -29700*A^2*B^2-59400*A^2*B*C-29700*A^2*C^2-7425*A*B^3-59400*A*B^2*C-59400*A*B*C ^2-7425*A*C^3-7425*B^3*C-29700*B^2*C^2-7425*B*C^3-2992*A*B-2992*A*C-2992*B*C-\ 1280)/(A+B+C) Hence the, 6, -th moment about the mean is 14 14 13 2 13 13 2 3 A B C (90 A B + 90 A C + 1350 A B + 2700 A B C + 1350 A C 12 3 12 2 12 2 12 3 + 9153 A B + 24192 A B C + 24192 A B C + 9153 A C 11 4 11 3 11 2 2 11 3 + 37089 A B + 109449 A B C + 144720 A B C + 109449 A B C 11 4 10 5 10 4 10 3 2 + 37089 A C + 100080 A B + 299430 A B C + 457686 A B C 10 2 3 10 4 10 5 9 6 + 457686 A B C + 299430 A B C + 100080 A C + 189684 A B 9 5 9 4 2 9 3 3 9 2 4 + 549054 A B C + 1016136 A B C + 1313532 A B C + 1016136 A B C 9 5 9 6 8 7 8 6 + 549054 A B C + 189684 A C + 259578 A B + 740520 A B C 8 5 2 8 4 3 8 3 4 + 1904562 A B C + 3063357 A B C + 3063357 A B C 8 2 5 8 6 8 7 7 8 + 1904562 A B C + 740520 A B C + 259578 A C + 259578 A B 7 7 7 6 2 7 5 3 7 4 4 + 806058 A B C + 2782890 A B C + 3658743 A B C + 2844666 A B C 7 3 5 7 2 6 7 7 7 8 + 3658743 A B C + 2782890 A B C + 806058 A B C + 259578 A C 6 9 6 8 6 7 2 6 6 3 + 189684 A B + 740520 A B C + 2782890 A B C + 3103056 A B C 6 5 4 6 4 5 6 3 6 + 9331740 A B C + 9331740 A B C + 3103056 A B C 6 2 7 6 8 6 9 5 10 + 2782890 A B C + 740520 A B C + 189684 A C + 100080 A B 5 9 5 8 2 5 7 3 5 6 4 + 549054 A B C + 1904562 A B C + 3658743 A B C + 9331740 A B C 5 5 5 5 4 6 5 3 7 - 2602908 A B C + 9331740 A B C + 3658743 A B C 5 2 8 5 9 5 10 4 11 + 1904562 A B C + 549054 A B C + 100080 A C + 37089 A B 4 10 4 9 2 4 8 3 4 7 4 + 299430 A B C + 1016136 A B C + 3063357 A B C + 2844666 A B C 4 6 5 4 5 6 4 4 7 + 9331740 A B C + 9331740 A B C + 2844666 A B C 4 3 8 4 2 9 4 10 4 11 + 3063357 A B C + 1016136 A B C + 299430 A B C + 37089 A C 3 12 3 11 3 10 2 3 9 3 + 9153 A B + 109449 A B C + 457686 A B C + 1313532 A B C 3 8 4 3 7 5 3 6 6 + 3063357 A B C + 3658743 A B C + 3103056 A B C 3 5 7 3 4 8 3 3 9 + 3658743 A B C + 3063357 A B C + 1313532 A B C 3 2 10 3 11 3 12 2 13 + 457686 A B C + 109449 A B C + 9153 A C + 1350 A B 2 12 2 11 2 2 10 3 2 9 4 + 24192 A B C + 144720 A B C + 457686 A B C + 1016136 A B C 2 8 5 2 7 6 2 6 7 + 1904562 A B C + 2782890 A B C + 2782890 A B C 2 5 8 2 4 9 2 3 10 + 1904562 A B C + 1016136 A B C + 457686 A B C 2 2 11 2 12 2 13 14 + 144720 A B C + 24192 A B C + 1350 A C + 90 A B 13 12 2 11 3 10 4 + 2700 A B C + 24192 A B C + 109449 A B C + 299430 A B C 9 5 8 6 7 7 6 8 + 549054 A B C + 740520 A B C + 806058 A B C + 740520 A B C 5 9 4 10 3 11 2 12 + 549054 A B C + 299430 A B C + 109449 A B C + 24192 A B C 13 14 14 13 2 12 3 + 2700 A B C + 90 A C + 90 B C + 1350 B C + 9153 B C 11 4 10 5 9 6 8 7 + 37089 B C + 100080 B C + 189684 B C + 259578 B C 7 8 6 9 5 10 4 11 3 12 + 259578 B C + 189684 B C + 100080 B C + 37089 B C + 9153 B C 2 13 14 12 12 11 2 + 1350 B C + 90 B C - 1485 A B - 1485 A C - 19305 A B 11 11 2 10 3 10 2 - 38610 A B C - 19305 A C - 110880 A B - 290070 A B C 10 2 10 3 9 4 9 3 - 290070 A B C - 110880 A C - 370260 A B - 1060290 A B C 9 2 2 9 3 9 4 8 5 - 1380060 A B C - 1060290 A B C - 370260 A C - 796455 A B 8 4 8 3 2 8 2 3 8 4 - 2268585 A B C - 3412530 A B C - 3412530 A B C - 2268585 A B C 8 5 7 6 7 5 7 4 2 - 796455 A C - 1156815 A B - 3261060 A B C - 6603795 A B C 7 3 3 7 2 4 7 5 7 6 - 8999100 A B C - 6603795 A B C - 3261060 A B C - 1156815 A C 6 7 6 6 6 5 2 6 4 3 - 1156815 A B - 3599640 A B C - 10010880 A B C - 13569930 A B C 6 3 4 6 2 5 6 6 6 7 - 13569930 A B C - 10010880 A B C - 3599640 A B C - 1156815 A C 5 8 5 7 5 6 2 5 5 3 - 796455 A B - 3261060 A B C - 10010880 A B C - 14067900 A B C 5 4 4 5 3 5 5 2 6 - 19459440 A B C - 14067900 A B C - 10010880 A B C 5 7 5 8 4 9 4 8 - 3261060 A B C - 796455 A C - 370260 A B - 2268585 A B C 4 7 2 4 6 3 4 5 4 - 6603795 A B C - 13569930 A B C - 19459440 A B C 4 4 5 4 3 6 4 2 7 - 19459440 A B C - 13569930 A B C - 6603795 A B C 4 8 4 9 3 10 3 9 - 2268585 A B C - 370260 A C - 110880 A B - 1060290 A B C 3 8 2 3 7 3 3 6 4 - 3412530 A B C - 8999100 A B C - 13569930 A B C 3 5 5 3 4 6 3 3 7 - 14067900 A B C - 13569930 A B C - 8999100 A B C 3 2 8 3 9 3 10 2 11 - 3412530 A B C - 1060290 A B C - 110880 A C - 19305 A B 2 10 2 9 2 2 8 3 2 7 4 - 290070 A B C - 1380060 A B C - 3412530 A B C - 6603795 A B C 2 6 5 2 5 6 2 4 7 - 10010880 A B C - 10010880 A B C - 6603795 A B C 2 3 8 2 2 9 2 10 2 11 - 3412530 A B C - 1380060 A B C - 290070 A B C - 19305 A C 12 11 10 2 9 3 - 1485 A B - 38610 A B C - 290070 A B C - 1060290 A B C 8 4 7 5 6 6 5 7 - 2268585 A B C - 3261060 A B C - 3599640 A B C - 3261060 A B C 4 8 3 9 2 10 11 - 2268585 A B C - 1060290 A B C - 290070 A B C - 38610 A B C 12 12 11 2 10 3 9 4 - 1485 A C - 1485 B C - 19305 B C - 110880 B C - 370260 B C 8 5 7 6 6 7 5 8 - 796455 B C - 1156815 B C - 1156815 B C - 796455 B C 4 9 3 10 2 11 12 10 - 370260 B C - 110880 B C - 19305 B C - 1485 B C + 6996 A B 10 9 2 9 9 2 8 3 + 6996 A C + 76956 A B + 153912 A B C + 76956 A C + 360294 A B 8 2 8 2 8 3 7 4 + 947364 A B C + 947364 A B C + 360294 A C + 933966 A B 7 3 7 2 2 7 3 7 4 + 2692206 A B C + 3516480 A B C + 2692206 A B C + 933966 A C 6 5 6 4 6 3 2 6 2 3 + 1476156 A B + 4530240 A B C + 7381572 A B C + 7381572 A B C 6 4 6 5 5 6 5 5 + 4530240 A B C + 1476156 A C + 1476156 A B + 5276964 A B C 5 4 2 5 3 3 5 2 4 + 10501260 A B C + 13677510 A B C + 10501260 A B C 5 5 5 6 4 7 4 6 + 5276964 A B C + 1476156 A C + 933966 A B + 4530240 A B C 4 5 2 4 4 3 4 3 4 + 10501260 A B C + 17255700 A B C + 17255700 A B C 4 2 5 4 6 4 7 3 8 + 10501260 A B C + 4530240 A B C + 933966 A C + 360294 A B 3 7 3 6 2 3 5 3 + 2692206 A B C + 7381572 A B C + 13677510 A B C 3 4 4 3 3 5 3 2 6 + 17255700 A B C + 13677510 A B C + 7381572 A B C 3 7 3 8 2 9 2 8 + 2692206 A B C + 360294 A C + 76956 A B + 947364 A B C 2 7 2 2 6 3 2 5 4 + 3516480 A B C + 7381572 A B C + 10501260 A B C 2 4 5 2 3 6 2 2 7 2 8 + 10501260 A B C + 7381572 A B C + 3516480 A B C + 947364 A B C 2 9 10 9 8 2 + 76956 A C + 6996 A B + 153912 A B C + 947364 A B C 7 3 6 4 5 5 4 6 + 2692206 A B C + 4530240 A B C + 5276964 A B C + 4530240 A B C 3 7 2 8 9 10 + 2692206 A B C + 947364 A B C + 153912 A B C + 6996 A C 10 9 2 8 3 7 4 6 5 + 6996 B C + 76956 B C + 360294 B C + 933966 B C + 1476156 B C 5 6 4 7 3 8 2 9 10 + 1476156 B C + 933966 B C + 360294 B C + 76956 B C + 6996 B C 8 8 7 2 7 7 2 - 7425 A B - 7425 A C - 66825 A B - 133650 A B C - 66825 A C 6 3 6 2 6 2 6 3 - 230175 A B - 680130 A B C - 680130 A B C - 230175 A C 5 4 5 3 5 2 2 5 3 - 408375 A B - 1636470 A B C - 2233440 A B C - 1636470 A B C 5 4 4 5 4 4 4 3 2 - 408375 A C - 408375 A B - 2165130 A B C - 3767445 A B C 4 2 3 4 4 4 5 3 6 - 3767445 A B C - 2165130 A B C - 408375 A C - 230175 A B 3 5 3 4 2 3 3 3 3 2 4 - 1636470 A B C - 3767445 A B C - 4722300 A B C - 3767445 A B C 3 5 3 6 2 7 2 6 - 1636470 A B C - 230175 A C - 66825 A B - 680130 A B C 2 5 2 2 4 3 2 3 4 - 2233440 A B C - 3767445 A B C - 3767445 A B C 2 2 5 2 6 2 7 8 - 2233440 A B C - 680130 A B C - 66825 A C - 7425 A B 7 6 2 5 3 4 4 - 133650 A B C - 680130 A B C - 1636470 A B C - 2165130 A B C 3 5 2 6 7 8 8 - 1636470 A B C - 680130 A B C - 133650 A B C - 7425 A C - 7425 B C 7 2 6 3 5 4 4 5 3 6 - 66825 B C - 230175 B C - 408375 B C - 408375 B C - 230175 B C 2 7 8 6 6 5 2 - 66825 B C - 7425 B C - 2992 A B - 2992 A C - 14960 A B 5 5 2 4 3 4 2 - 39248 A B C - 14960 A C - 29920 A B - 130064 A B C 4 2 4 3 3 4 3 3 - 130064 A B C - 29920 A C - 29920 A B - 187616 A B C 3 2 2 3 3 3 4 2 5 - 315392 A B C - 187616 A B C - 29920 A C - 14960 A B 2 4 2 3 2 2 2 3 2 4 - 130064 A B C - 315392 A B C - 315392 A B C - 130064 A B C 2 5 6 5 4 2 3 3 - 14960 A C - 2992 A B - 39248 A B C - 130064 A B C - 187616 A B C 2 4 5 6 6 5 2 - 130064 A B C - 39248 A B C - 2992 A C - 2992 B C - 14960 B C 4 3 3 4 2 5 6 5 - 29920 B C - 29920 B C - 14960 B C - 2992 B C - 1280 A 4 4 3 2 3 3 2 - 6400 A B - 6400 A C - 12800 A B - 25600 A B C - 12800 A C 2 3 2 2 2 2 2 3 4 - 12800 A B - 38400 A B C - 38400 A B C - 12800 A C - 6400 A B 3 2 2 3 4 5 - 25600 A B C - 38400 A B C - 25600 A B C - 6400 A C - 1280 B 4 3 2 2 3 4 5 / - 6400 B C - 12800 B C - 12800 B C - 6400 B C - 1280 C ) / (24640 / 6 (A + B + C) ) and in Maple notation 3/24640*A*B*C*(90*A^14*B+90*A^14*C+1350*A^13*B^2+2700*A^13*B*C+1350*A^13*C^2+ 9153*A^12*B^3+24192*A^12*B^2*C+24192*A^12*B*C^2+9153*A^12*C^3+37089*A^11*B^4+ 109449*A^11*B^3*C+144720*A^11*B^2*C^2+109449*A^11*B*C^3+37089*A^11*C^4+100080*A ^10*B^5+299430*A^10*B^4*C+457686*A^10*B^3*C^2+457686*A^10*B^2*C^3+299430*A^10*B *C^4+100080*A^10*C^5+189684*A^9*B^6+549054*A^9*B^5*C+1016136*A^9*B^4*C^2+ 1313532*A^9*B^3*C^3+1016136*A^9*B^2*C^4+549054*A^9*B*C^5+189684*A^9*C^6+259578* A^8*B^7+740520*A^8*B^6*C+1904562*A^8*B^5*C^2+3063357*A^8*B^4*C^3+3063357*A^8*B^ 3*C^4+1904562*A^8*B^2*C^5+740520*A^8*B*C^6+259578*A^8*C^7+259578*A^7*B^8+806058 *A^7*B^7*C+2782890*A^7*B^6*C^2+3658743*A^7*B^5*C^3+2844666*A^7*B^4*C^4+3658743* A^7*B^3*C^5+2782890*A^7*B^2*C^6+806058*A^7*B*C^7+259578*A^7*C^8+189684*A^6*B^9+ 740520*A^6*B^8*C+2782890*A^6*B^7*C^2+3103056*A^6*B^6*C^3+9331740*A^6*B^5*C^4+ 9331740*A^6*B^4*C^5+3103056*A^6*B^3*C^6+2782890*A^6*B^2*C^7+740520*A^6*B*C^8+ 189684*A^6*C^9+100080*A^5*B^10+549054*A^5*B^9*C+1904562*A^5*B^8*C^2+3658743*A^5 *B^7*C^3+9331740*A^5*B^6*C^4-2602908*A^5*B^5*C^5+9331740*A^5*B^4*C^6+3658743*A^ 5*B^3*C^7+1904562*A^5*B^2*C^8+549054*A^5*B*C^9+100080*A^5*C^10+37089*A^4*B^11+ 299430*A^4*B^10*C+1016136*A^4*B^9*C^2+3063357*A^4*B^8*C^3+2844666*A^4*B^7*C^4+ 9331740*A^4*B^6*C^5+9331740*A^4*B^5*C^6+2844666*A^4*B^4*C^7+3063357*A^4*B^3*C^8 +1016136*A^4*B^2*C^9+299430*A^4*B*C^10+37089*A^4*C^11+9153*A^3*B^12+109449*A^3* B^11*C+457686*A^3*B^10*C^2+1313532*A^3*B^9*C^3+3063357*A^3*B^8*C^4+3658743*A^3* B^7*C^5+3103056*A^3*B^6*C^6+3658743*A^3*B^5*C^7+3063357*A^3*B^4*C^8+1313532*A^3 *B^3*C^9+457686*A^3*B^2*C^10+109449*A^3*B*C^11+9153*A^3*C^12+1350*A^2*B^13+ 24192*A^2*B^12*C+144720*A^2*B^11*C^2+457686*A^2*B^10*C^3+1016136*A^2*B^9*C^4+ 1904562*A^2*B^8*C^5+2782890*A^2*B^7*C^6+2782890*A^2*B^6*C^7+1904562*A^2*B^5*C^8 +1016136*A^2*B^4*C^9+457686*A^2*B^3*C^10+144720*A^2*B^2*C^11+24192*A^2*B*C^12+ 1350*A^2*C^13+90*A*B^14+2700*A*B^13*C+24192*A*B^12*C^2+109449*A*B^11*C^3+299430 *A*B^10*C^4+549054*A*B^9*C^5+740520*A*B^8*C^6+806058*A*B^7*C^7+740520*A*B^6*C^8 +549054*A*B^5*C^9+299430*A*B^4*C^10+109449*A*B^3*C^11+24192*A*B^2*C^12+2700*A*B *C^13+90*A*C^14+90*B^14*C+1350*B^13*C^2+9153*B^12*C^3+37089*B^11*C^4+100080*B^ 10*C^5+189684*B^9*C^6+259578*B^8*C^7+259578*B^7*C^8+189684*B^6*C^9+100080*B^5*C ^10+37089*B^4*C^11+9153*B^3*C^12+1350*B^2*C^13+90*B*C^14-1485*A^12*B-1485*A^12* C-19305*A^11*B^2-38610*A^11*B*C-19305*A^11*C^2-110880*A^10*B^3-290070*A^10*B^2* C-290070*A^10*B*C^2-110880*A^10*C^3-370260*A^9*B^4-1060290*A^9*B^3*C-1380060*A^ 9*B^2*C^2-1060290*A^9*B*C^3-370260*A^9*C^4-796455*A^8*B^5-2268585*A^8*B^4*C-\ 3412530*A^8*B^3*C^2-3412530*A^8*B^2*C^3-2268585*A^8*B*C^4-796455*A^8*C^5-\ 1156815*A^7*B^6-3261060*A^7*B^5*C-6603795*A^7*B^4*C^2-8999100*A^7*B^3*C^3-\ 6603795*A^7*B^2*C^4-3261060*A^7*B*C^5-1156815*A^7*C^6-1156815*A^6*B^7-3599640*A ^6*B^6*C-10010880*A^6*B^5*C^2-13569930*A^6*B^4*C^3-13569930*A^6*B^3*C^4-\ 10010880*A^6*B^2*C^5-3599640*A^6*B*C^6-1156815*A^6*C^7-796455*A^5*B^8-3261060*A ^5*B^7*C-10010880*A^5*B^6*C^2-14067900*A^5*B^5*C^3-19459440*A^5*B^4*C^4-\ 14067900*A^5*B^3*C^5-10010880*A^5*B^2*C^6-3261060*A^5*B*C^7-796455*A^5*C^8-\ 370260*A^4*B^9-2268585*A^4*B^8*C-6603795*A^4*B^7*C^2-13569930*A^4*B^6*C^3-\ 19459440*A^4*B^5*C^4-19459440*A^4*B^4*C^5-13569930*A^4*B^3*C^6-6603795*A^4*B^2* C^7-2268585*A^4*B*C^8-370260*A^4*C^9-110880*A^3*B^10-1060290*A^3*B^9*C-3412530* A^3*B^8*C^2-8999100*A^3*B^7*C^3-13569930*A^3*B^6*C^4-14067900*A^3*B^5*C^5-\ 13569930*A^3*B^4*C^6-8999100*A^3*B^3*C^7-3412530*A^3*B^2*C^8-1060290*A^3*B*C^9-\ 110880*A^3*C^10-19305*A^2*B^11-290070*A^2*B^10*C-1380060*A^2*B^9*C^2-3412530*A^ 2*B^8*C^3-6603795*A^2*B^7*C^4-10010880*A^2*B^6*C^5-10010880*A^2*B^5*C^6-6603795 *A^2*B^4*C^7-3412530*A^2*B^3*C^8-1380060*A^2*B^2*C^9-290070*A^2*B*C^10-19305*A^ 2*C^11-1485*A*B^12-38610*A*B^11*C-290070*A*B^10*C^2-1060290*A*B^9*C^3-2268585*A *B^8*C^4-3261060*A*B^7*C^5-3599640*A*B^6*C^6-3261060*A*B^5*C^7-2268585*A*B^4*C^ 8-1060290*A*B^3*C^9-290070*A*B^2*C^10-38610*A*B*C^11-1485*A*C^12-1485*B^12*C-\ 19305*B^11*C^2-110880*B^10*C^3-370260*B^9*C^4-796455*B^8*C^5-1156815*B^7*C^6-\ 1156815*B^6*C^7-796455*B^5*C^8-370260*B^4*C^9-110880*B^3*C^10-19305*B^2*C^11-\ 1485*B*C^12+6996*A^10*B+6996*A^10*C+76956*A^9*B^2+153912*A^9*B*C+76956*A^9*C^2+ 360294*A^8*B^3+947364*A^8*B^2*C+947364*A^8*B*C^2+360294*A^8*C^3+933966*A^7*B^4+ 2692206*A^7*B^3*C+3516480*A^7*B^2*C^2+2692206*A^7*B*C^3+933966*A^7*C^4+1476156* A^6*B^5+4530240*A^6*B^4*C+7381572*A^6*B^3*C^2+7381572*A^6*B^2*C^3+4530240*A^6*B *C^4+1476156*A^6*C^5+1476156*A^5*B^6+5276964*A^5*B^5*C+10501260*A^5*B^4*C^2+ 13677510*A^5*B^3*C^3+10501260*A^5*B^2*C^4+5276964*A^5*B*C^5+1476156*A^5*C^6+ 933966*A^4*B^7+4530240*A^4*B^6*C+10501260*A^4*B^5*C^2+17255700*A^4*B^4*C^3+ 17255700*A^4*B^3*C^4+10501260*A^4*B^2*C^5+4530240*A^4*B*C^6+933966*A^4*C^7+ 360294*A^3*B^8+2692206*A^3*B^7*C+7381572*A^3*B^6*C^2+13677510*A^3*B^5*C^3+ 17255700*A^3*B^4*C^4+13677510*A^3*B^3*C^5+7381572*A^3*B^2*C^6+2692206*A^3*B*C^7 +360294*A^3*C^8+76956*A^2*B^9+947364*A^2*B^8*C+3516480*A^2*B^7*C^2+7381572*A^2* B^6*C^3+10501260*A^2*B^5*C^4+10501260*A^2*B^4*C^5+7381572*A^2*B^3*C^6+3516480*A ^2*B^2*C^7+947364*A^2*B*C^8+76956*A^2*C^9+6996*A*B^10+153912*A*B^9*C+947364*A*B ^8*C^2+2692206*A*B^7*C^3+4530240*A*B^6*C^4+5276964*A*B^5*C^5+4530240*A*B^4*C^6+ 2692206*A*B^3*C^7+947364*A*B^2*C^8+153912*A*B*C^9+6996*A*C^10+6996*B^10*C+76956 *B^9*C^2+360294*B^8*C^3+933966*B^7*C^4+1476156*B^6*C^5+1476156*B^5*C^6+933966*B ^4*C^7+360294*B^3*C^8+76956*B^2*C^9+6996*B*C^10-7425*A^8*B-7425*A^8*C-66825*A^7 *B^2-133650*A^7*B*C-66825*A^7*C^2-230175*A^6*B^3-680130*A^6*B^2*C-680130*A^6*B* C^2-230175*A^6*C^3-408375*A^5*B^4-1636470*A^5*B^3*C-2233440*A^5*B^2*C^2-1636470 *A^5*B*C^3-408375*A^5*C^4-408375*A^4*B^5-2165130*A^4*B^4*C-3767445*A^4*B^3*C^2-\ 3767445*A^4*B^2*C^3-2165130*A^4*B*C^4-408375*A^4*C^5-230175*A^3*B^6-1636470*A^3 *B^5*C-3767445*A^3*B^4*C^2-4722300*A^3*B^3*C^3-3767445*A^3*B^2*C^4-1636470*A^3* B*C^5-230175*A^3*C^6-66825*A^2*B^7-680130*A^2*B^6*C-2233440*A^2*B^5*C^2-3767445 *A^2*B^4*C^3-3767445*A^2*B^3*C^4-2233440*A^2*B^2*C^5-680130*A^2*B*C^6-66825*A^2 *C^7-7425*A*B^8-133650*A*B^7*C-680130*A*B^6*C^2-1636470*A*B^5*C^3-2165130*A*B^4 *C^4-1636470*A*B^3*C^5-680130*A*B^2*C^6-133650*A*B*C^7-7425*A*C^8-7425*B^8*C-\ 66825*B^7*C^2-230175*B^6*C^3-408375*B^5*C^4-408375*B^4*C^5-230175*B^3*C^6-66825 *B^2*C^7-7425*B*C^8-2992*A^6*B-2992*A^6*C-14960*A^5*B^2-39248*A^5*B*C-14960*A^5 *C^2-29920*A^4*B^3-130064*A^4*B^2*C-130064*A^4*B*C^2-29920*A^4*C^3-29920*A^3*B^ 4-187616*A^3*B^3*C-315392*A^3*B^2*C^2-187616*A^3*B*C^3-29920*A^3*C^4-14960*A^2* B^5-130064*A^2*B^4*C-315392*A^2*B^3*C^2-315392*A^2*B^2*C^3-130064*A^2*B*C^4-\ 14960*A^2*C^5-2992*A*B^6-39248*A*B^5*C-130064*A*B^4*C^2-187616*A*B^3*C^3-130064 *A*B^2*C^4-39248*A*B*C^5-2992*A*C^6-2992*B^6*C-14960*B^5*C^2-29920*B^4*C^3-\ 29920*B^3*C^4-14960*B^2*C^5-2992*B*C^6-1280*A^5-6400*A^4*B-6400*A^4*C-12800*A^3 *B^2-25600*A^3*B*C-12800*A^3*C^2-12800*A^2*B^3-38400*A^2*B^2*C-38400*A^2*B*C^2-\ 12800*A^2*C^3-6400*A*B^4-25600*A*B^3*C-38400*A*B^2*C^2-25600*A*B*C^3-6400*A*C^4 -1280*B^5-6400*B^4*C-12800*B^3*C^2-12800*B^2*C^3-6400*B*C^4-1280*C^5)/(A+B+C)^6 Hence the scaled, 6, -th moment about the mean is 14 14 13 2 13 13 2 12 3 (90 A B + 90 A C + 1350 A B + 2700 A B C + 1350 A C + 9153 A B 12 2 12 2 12 3 11 4 + 24192 A B C + 24192 A B C + 9153 A C + 37089 A B 11 3 11 2 2 11 3 11 4 + 109449 A B C + 144720 A B C + 109449 A B C + 37089 A C 10 5 10 4 10 3 2 10 2 3 + 100080 A B + 299430 A B C + 457686 A B C + 457686 A B C 10 4 10 5 9 6 9 5 + 299430 A B C + 100080 A C + 189684 A B + 549054 A B C 9 4 2 9 3 3 9 2 4 9 5 + 1016136 A B C + 1313532 A B C + 1016136 A B C + 549054 A B C 9 6 8 7 8 6 8 5 2 + 189684 A C + 259578 A B + 740520 A B C + 1904562 A B C 8 4 3 8 3 4 8 2 5 8 6 + 3063357 A B C + 3063357 A B C + 1904562 A B C + 740520 A B C 8 7 7 8 7 7 7 6 2 + 259578 A C + 259578 A B + 806058 A B C + 2782890 A B C 7 5 3 7 4 4 7 3 5 + 3658743 A B C + 2844666 A B C + 3658743 A B C 7 2 6 7 7 7 8 6 9 + 2782890 A B C + 806058 A B C + 259578 A C + 189684 A B 6 8 6 7 2 6 6 3 6 5 4 + 740520 A B C + 2782890 A B C + 3103056 A B C + 9331740 A B C 6 4 5 6 3 6 6 2 7 6 8 + 9331740 A B C + 3103056 A B C + 2782890 A B C + 740520 A B C 6 9 5 10 5 9 5 8 2 + 189684 A C + 100080 A B + 549054 A B C + 1904562 A B C 5 7 3 5 6 4 5 5 5 + 3658743 A B C + 9331740 A B C - 2602908 A B C 5 4 6 5 3 7 5 2 8 5 9 + 9331740 A B C + 3658743 A B C + 1904562 A B C + 549054 A B C 5 10 4 11 4 10 4 9 2 + 100080 A C + 37089 A B + 299430 A B C + 1016136 A B C 4 8 3 4 7 4 4 6 5 + 3063357 A B C + 2844666 A B C + 9331740 A B C 4 5 6 4 4 7 4 3 8 + 9331740 A B C + 2844666 A B C + 3063357 A B C 4 2 9 4 10 4 11 3 12 + 1016136 A B C + 299430 A B C + 37089 A C + 9153 A B 3 11 3 10 2 3 9 3 3 8 4 + 109449 A B C + 457686 A B C + 1313532 A B C + 3063357 A B C 3 7 5 3 6 6 3 5 7 + 3658743 A B C + 3103056 A B C + 3658743 A B C 3 4 8 3 3 9 3 2 10 3 11 + 3063357 A B C + 1313532 A B C + 457686 A B C + 109449 A B C 3 12 2 13 2 12 2 11 2 + 9153 A C + 1350 A B + 24192 A B C + 144720 A B C 2 10 3 2 9 4 2 8 5 + 457686 A B C + 1016136 A B C + 1904562 A B C 2 7 6 2 6 7 2 5 8 + 2782890 A B C + 2782890 A B C + 1904562 A B C 2 4 9 2 3 10 2 2 11 2 12 + 1016136 A B C + 457686 A B C + 144720 A B C + 24192 A B C 2 13 14 13 12 2 11 3 + 1350 A C + 90 A B + 2700 A B C + 24192 A B C + 109449 A B C 10 4 9 5 8 6 7 7 + 299430 A B C + 549054 A B C + 740520 A B C + 806058 A B C 6 8 5 9 4 10 3 11 + 740520 A B C + 549054 A B C + 299430 A B C + 109449 A B C 2 12 13 14 14 13 2 + 24192 A B C + 2700 A B C + 90 A C + 90 B C + 1350 B C 12 3 11 4 10 5 9 6 8 7 + 9153 B C + 37089 B C + 100080 B C + 189684 B C + 259578 B C 7 8 6 9 5 10 4 11 3 12 + 259578 B C + 189684 B C + 100080 B C + 37089 B C + 9153 B C 2 13 14 12 12 11 2 + 1350 B C + 90 B C - 1485 A B - 1485 A C - 19305 A B 11 11 2 10 3 10 2 - 38610 A B C - 19305 A C - 110880 A B - 290070 A B C 10 2 10 3 9 4 9 3 - 290070 A B C - 110880 A C - 370260 A B - 1060290 A B C 9 2 2 9 3 9 4 8 5 - 1380060 A B C - 1060290 A B C - 370260 A C - 796455 A B 8 4 8 3 2 8 2 3 8 4 - 2268585 A B C - 3412530 A B C - 3412530 A B C - 2268585 A B C 8 5 7 6 7 5 7 4 2 - 796455 A C - 1156815 A B - 3261060 A B C - 6603795 A B C 7 3 3 7 2 4 7 5 7 6 - 8999100 A B C - 6603795 A B C - 3261060 A B C - 1156815 A C 6 7 6 6 6 5 2 6 4 3 - 1156815 A B - 3599640 A B C - 10010880 A B C - 13569930 A B C 6 3 4 6 2 5 6 6 6 7 - 13569930 A B C - 10010880 A B C - 3599640 A B C - 1156815 A C 5 8 5 7 5 6 2 5 5 3 - 796455 A B - 3261060 A B C - 10010880 A B C - 14067900 A B C 5 4 4 5 3 5 5 2 6 - 19459440 A B C - 14067900 A B C - 10010880 A B C 5 7 5 8 4 9 4 8 - 3261060 A B C - 796455 A C - 370260 A B - 2268585 A B C 4 7 2 4 6 3 4 5 4 - 6603795 A B C - 13569930 A B C - 19459440 A B C 4 4 5 4 3 6 4 2 7 - 19459440 A B C - 13569930 A B C - 6603795 A B C 4 8 4 9 3 10 3 9 - 2268585 A B C - 370260 A C - 110880 A B - 1060290 A B C 3 8 2 3 7 3 3 6 4 - 3412530 A B C - 8999100 A B C - 13569930 A B C 3 5 5 3 4 6 3 3 7 - 14067900 A B C - 13569930 A B C - 8999100 A B C 3 2 8 3 9 3 10 2 11 - 3412530 A B C - 1060290 A B C - 110880 A C - 19305 A B 2 10 2 9 2 2 8 3 2 7 4 - 290070 A B C - 1380060 A B C - 3412530 A B C - 6603795 A B C 2 6 5 2 5 6 2 4 7 - 10010880 A B C - 10010880 A B C - 6603795 A B C 2 3 8 2 2 9 2 10 2 11 - 3412530 A B C - 1380060 A B C - 290070 A B C - 19305 A C 12 11 10 2 9 3 - 1485 A B - 38610 A B C - 290070 A B C - 1060290 A B C 8 4 7 5 6 6 5 7 - 2268585 A B C - 3261060 A B C - 3599640 A B C - 3261060 A B C 4 8 3 9 2 10 11 - 2268585 A B C - 1060290 A B C - 290070 A B C - 38610 A B C 12 12 11 2 10 3 9 4 - 1485 A C - 1485 B C - 19305 B C - 110880 B C - 370260 B C 8 5 7 6 6 7 5 8 - 796455 B C - 1156815 B C - 1156815 B C - 796455 B C 4 9 3 10 2 11 12 10 - 370260 B C - 110880 B C - 19305 B C - 1485 B C + 6996 A B 10 9 2 9 9 2 8 3 + 6996 A C + 76956 A B + 153912 A B C + 76956 A C + 360294 A B 8 2 8 2 8 3 7 4 + 947364 A B C + 947364 A B C + 360294 A C + 933966 A B 7 3 7 2 2 7 3 7 4 + 2692206 A B C + 3516480 A B C + 2692206 A B C + 933966 A C 6 5 6 4 6 3 2 6 2 3 + 1476156 A B + 4530240 A B C + 7381572 A B C + 7381572 A B C 6 4 6 5 5 6 5 5 + 4530240 A B C + 1476156 A C + 1476156 A B + 5276964 A B C 5 4 2 5 3 3 5 2 4 + 10501260 A B C + 13677510 A B C + 10501260 A B C 5 5 5 6 4 7 4 6 + 5276964 A B C + 1476156 A C + 933966 A B + 4530240 A B C 4 5 2 4 4 3 4 3 4 + 10501260 A B C + 17255700 A B C + 17255700 A B C 4 2 5 4 6 4 7 3 8 + 10501260 A B C + 4530240 A B C + 933966 A C + 360294 A B 3 7 3 6 2 3 5 3 + 2692206 A B C + 7381572 A B C + 13677510 A B C 3 4 4 3 3 5 3 2 6 + 17255700 A B C + 13677510 A B C + 7381572 A B C 3 7 3 8 2 9 2 8 + 2692206 A B C + 360294 A C + 76956 A B + 947364 A B C 2 7 2 2 6 3 2 5 4 + 3516480 A B C + 7381572 A B C + 10501260 A B C 2 4 5 2 3 6 2 2 7 2 8 + 10501260 A B C + 7381572 A B C + 3516480 A B C + 947364 A B C 2 9 10 9 8 2 + 76956 A C + 6996 A B + 153912 A B C + 947364 A B C 7 3 6 4 5 5 4 6 + 2692206 A B C + 4530240 A B C + 5276964 A B C + 4530240 A B C 3 7 2 8 9 10 + 2692206 A B C + 947364 A B C + 153912 A B C + 6996 A C 10 9 2 8 3 7 4 6 5 + 6996 B C + 76956 B C + 360294 B C + 933966 B C + 1476156 B C 5 6 4 7 3 8 2 9 10 + 1476156 B C + 933966 B C + 360294 B C + 76956 B C + 6996 B C 8 8 7 2 7 7 2 - 7425 A B - 7425 A C - 66825 A B - 133650 A B C - 66825 A C 6 3 6 2 6 2 6 3 - 230175 A B - 680130 A B C - 680130 A B C - 230175 A C 5 4 5 3 5 2 2 5 3 - 408375 A B - 1636470 A B C - 2233440 A B C - 1636470 A B C 5 4 4 5 4 4 4 3 2 - 408375 A C - 408375 A B - 2165130 A B C - 3767445 A B C 4 2 3 4 4 4 5 3 6 - 3767445 A B C - 2165130 A B C - 408375 A C - 230175 A B 3 5 3 4 2 3 3 3 3 2 4 - 1636470 A B C - 3767445 A B C - 4722300 A B C - 3767445 A B C 3 5 3 6 2 7 2 6 - 1636470 A B C - 230175 A C - 66825 A B - 680130 A B C 2 5 2 2 4 3 2 3 4 - 2233440 A B C - 3767445 A B C - 3767445 A B C 2 2 5 2 6 2 7 8 - 2233440 A B C - 680130 A B C - 66825 A C - 7425 A B 7 6 2 5 3 4 4 - 133650 A B C - 680130 A B C - 1636470 A B C - 2165130 A B C 3 5 2 6 7 8 8 - 1636470 A B C - 680130 A B C - 133650 A B C - 7425 A C - 7425 B C 7 2 6 3 5 4 4 5 3 6 - 66825 B C - 230175 B C - 408375 B C - 408375 B C - 230175 B C 2 7 8 6 6 5 2 - 66825 B C - 7425 B C - 2992 A B - 2992 A C - 14960 A B 5 5 2 4 3 4 2 - 39248 A B C - 14960 A C - 29920 A B - 130064 A B C 4 2 4 3 3 4 3 3 - 130064 A B C - 29920 A C - 29920 A B - 187616 A B C 3 2 2 3 3 3 4 2 5 - 315392 A B C - 187616 A B C - 29920 A C - 14960 A B 2 4 2 3 2 2 2 3 2 4 - 130064 A B C - 315392 A B C - 315392 A B C - 130064 A B C 2 5 6 5 4 2 3 3 - 14960 A C - 2992 A B - 39248 A B C - 130064 A B C - 187616 A B C 2 4 5 6 6 5 2 - 130064 A B C - 39248 A B C - 2992 A C - 2992 B C - 14960 B C 4 3 3 4 2 5 6 5 - 29920 B C - 29920 B C - 14960 B C - 2992 B C - 1280 A 4 4 3 2 3 3 2 - 6400 A B - 6400 A C - 12800 A B - 25600 A B C - 12800 A C 2 3 2 2 2 2 2 3 4 - 12800 A B - 38400 A B C - 38400 A B C - 12800 A C - 6400 A B 3 2 2 3 4 5 - 25600 A B C - 38400 A B C - 25600 A B C - 6400 A C - 1280 B 4 3 2 2 3 4 5 / - 6400 B C - 12800 B C - 12800 B C - 6400 B C - 1280 C ) / (27720 / 2 2 2 2 2 2 2 2 2 3 A B C (A B + A C + A B - 3 A B C + A C + B C + B C - A - B - C) ) and in Maple notation 1/27720/A^2/B^2/C^2*(90*A^14*B+90*A^14*C+1350*A^13*B^2+2700*A^13*B*C+1350*A^13* C^2+9153*A^12*B^3+24192*A^12*B^2*C+24192*A^12*B*C^2+9153*A^12*C^3+37089*A^11*B^ 4+109449*A^11*B^3*C+144720*A^11*B^2*C^2+109449*A^11*B*C^3+37089*A^11*C^4+100080 *A^10*B^5+299430*A^10*B^4*C+457686*A^10*B^3*C^2+457686*A^10*B^2*C^3+299430*A^10 *B*C^4+100080*A^10*C^5+189684*A^9*B^6+549054*A^9*B^5*C+1016136*A^9*B^4*C^2+ 1313532*A^9*B^3*C^3+1016136*A^9*B^2*C^4+549054*A^9*B*C^5+189684*A^9*C^6+259578* A^8*B^7+740520*A^8*B^6*C+1904562*A^8*B^5*C^2+3063357*A^8*B^4*C^3+3063357*A^8*B^ 3*C^4+1904562*A^8*B^2*C^5+740520*A^8*B*C^6+259578*A^8*C^7+259578*A^7*B^8+806058 *A^7*B^7*C+2782890*A^7*B^6*C^2+3658743*A^7*B^5*C^3+2844666*A^7*B^4*C^4+3658743* A^7*B^3*C^5+2782890*A^7*B^2*C^6+806058*A^7*B*C^7+259578*A^7*C^8+189684*A^6*B^9+ 740520*A^6*B^8*C+2782890*A^6*B^7*C^2+3103056*A^6*B^6*C^3+9331740*A^6*B^5*C^4+ 9331740*A^6*B^4*C^5+3103056*A^6*B^3*C^6+2782890*A^6*B^2*C^7+740520*A^6*B*C^8+ 189684*A^6*C^9+100080*A^5*B^10+549054*A^5*B^9*C+1904562*A^5*B^8*C^2+3658743*A^5 *B^7*C^3+9331740*A^5*B^6*C^4-2602908*A^5*B^5*C^5+9331740*A^5*B^4*C^6+3658743*A^ 5*B^3*C^7+1904562*A^5*B^2*C^8+549054*A^5*B*C^9+100080*A^5*C^10+37089*A^4*B^11+ 299430*A^4*B^10*C+1016136*A^4*B^9*C^2+3063357*A^4*B^8*C^3+2844666*A^4*B^7*C^4+ 9331740*A^4*B^6*C^5+9331740*A^4*B^5*C^6+2844666*A^4*B^4*C^7+3063357*A^4*B^3*C^8 +1016136*A^4*B^2*C^9+299430*A^4*B*C^10+37089*A^4*C^11+9153*A^3*B^12+109449*A^3* B^11*C+457686*A^3*B^10*C^2+1313532*A^3*B^9*C^3+3063357*A^3*B^8*C^4+3658743*A^3* B^7*C^5+3103056*A^3*B^6*C^6+3658743*A^3*B^5*C^7+3063357*A^3*B^4*C^8+1313532*A^3 *B^3*C^9+457686*A^3*B^2*C^10+109449*A^3*B*C^11+9153*A^3*C^12+1350*A^2*B^13+ 24192*A^2*B^12*C+144720*A^2*B^11*C^2+457686*A^2*B^10*C^3+1016136*A^2*B^9*C^4+ 1904562*A^2*B^8*C^5+2782890*A^2*B^7*C^6+2782890*A^2*B^6*C^7+1904562*A^2*B^5*C^8 +1016136*A^2*B^4*C^9+457686*A^2*B^3*C^10+144720*A^2*B^2*C^11+24192*A^2*B*C^12+ 1350*A^2*C^13+90*A*B^14+2700*A*B^13*C+24192*A*B^12*C^2+109449*A*B^11*C^3+299430 *A*B^10*C^4+549054*A*B^9*C^5+740520*A*B^8*C^6+806058*A*B^7*C^7+740520*A*B^6*C^8 +549054*A*B^5*C^9+299430*A*B^4*C^10+109449*A*B^3*C^11+24192*A*B^2*C^12+2700*A*B *C^13+90*A*C^14+90*B^14*C+1350*B^13*C^2+9153*B^12*C^3+37089*B^11*C^4+100080*B^ 10*C^5+189684*B^9*C^6+259578*B^8*C^7+259578*B^7*C^8+189684*B^6*C^9+100080*B^5*C ^10+37089*B^4*C^11+9153*B^3*C^12+1350*B^2*C^13+90*B*C^14-1485*A^12*B-1485*A^12* C-19305*A^11*B^2-38610*A^11*B*C-19305*A^11*C^2-110880*A^10*B^3-290070*A^10*B^2* C-290070*A^10*B*C^2-110880*A^10*C^3-370260*A^9*B^4-1060290*A^9*B^3*C-1380060*A^ 9*B^2*C^2-1060290*A^9*B*C^3-370260*A^9*C^4-796455*A^8*B^5-2268585*A^8*B^4*C-\ 3412530*A^8*B^3*C^2-3412530*A^8*B^2*C^3-2268585*A^8*B*C^4-796455*A^8*C^5-\ 1156815*A^7*B^6-3261060*A^7*B^5*C-6603795*A^7*B^4*C^2-8999100*A^7*B^3*C^3-\ 6603795*A^7*B^2*C^4-3261060*A^7*B*C^5-1156815*A^7*C^6-1156815*A^6*B^7-3599640*A ^6*B^6*C-10010880*A^6*B^5*C^2-13569930*A^6*B^4*C^3-13569930*A^6*B^3*C^4-\ 10010880*A^6*B^2*C^5-3599640*A^6*B*C^6-1156815*A^6*C^7-796455*A^5*B^8-3261060*A ^5*B^7*C-10010880*A^5*B^6*C^2-14067900*A^5*B^5*C^3-19459440*A^5*B^4*C^4-\ 14067900*A^5*B^3*C^5-10010880*A^5*B^2*C^6-3261060*A^5*B*C^7-796455*A^5*C^8-\ 370260*A^4*B^9-2268585*A^4*B^8*C-6603795*A^4*B^7*C^2-13569930*A^4*B^6*C^3-\ 19459440*A^4*B^5*C^4-19459440*A^4*B^4*C^5-13569930*A^4*B^3*C^6-6603795*A^4*B^2* C^7-2268585*A^4*B*C^8-370260*A^4*C^9-110880*A^3*B^10-1060290*A^3*B^9*C-3412530* A^3*B^8*C^2-8999100*A^3*B^7*C^3-13569930*A^3*B^6*C^4-14067900*A^3*B^5*C^5-\ 13569930*A^3*B^4*C^6-8999100*A^3*B^3*C^7-3412530*A^3*B^2*C^8-1060290*A^3*B*C^9-\ 110880*A^3*C^10-19305*A^2*B^11-290070*A^2*B^10*C-1380060*A^2*B^9*C^2-3412530*A^ 2*B^8*C^3-6603795*A^2*B^7*C^4-10010880*A^2*B^6*C^5-10010880*A^2*B^5*C^6-6603795 *A^2*B^4*C^7-3412530*A^2*B^3*C^8-1380060*A^2*B^2*C^9-290070*A^2*B*C^10-19305*A^ 2*C^11-1485*A*B^12-38610*A*B^11*C-290070*A*B^10*C^2-1060290*A*B^9*C^3-2268585*A *B^8*C^4-3261060*A*B^7*C^5-3599640*A*B^6*C^6-3261060*A*B^5*C^7-2268585*A*B^4*C^ 8-1060290*A*B^3*C^9-290070*A*B^2*C^10-38610*A*B*C^11-1485*A*C^12-1485*B^12*C-\ 19305*B^11*C^2-110880*B^10*C^3-370260*B^9*C^4-796455*B^8*C^5-1156815*B^7*C^6-\ 1156815*B^6*C^7-796455*B^5*C^8-370260*B^4*C^9-110880*B^3*C^10-19305*B^2*C^11-\ 1485*B*C^12+6996*A^10*B+6996*A^10*C+76956*A^9*B^2+153912*A^9*B*C+76956*A^9*C^2+ 360294*A^8*B^3+947364*A^8*B^2*C+947364*A^8*B*C^2+360294*A^8*C^3+933966*A^7*B^4+ 2692206*A^7*B^3*C+3516480*A^7*B^2*C^2+2692206*A^7*B*C^3+933966*A^7*C^4+1476156* A^6*B^5+4530240*A^6*B^4*C+7381572*A^6*B^3*C^2+7381572*A^6*B^2*C^3+4530240*A^6*B *C^4+1476156*A^6*C^5+1476156*A^5*B^6+5276964*A^5*B^5*C+10501260*A^5*B^4*C^2+ 13677510*A^5*B^3*C^3+10501260*A^5*B^2*C^4+5276964*A^5*B*C^5+1476156*A^5*C^6+ 933966*A^4*B^7+4530240*A^4*B^6*C+10501260*A^4*B^5*C^2+17255700*A^4*B^4*C^3+ 17255700*A^4*B^3*C^4+10501260*A^4*B^2*C^5+4530240*A^4*B*C^6+933966*A^4*C^7+ 360294*A^3*B^8+2692206*A^3*B^7*C+7381572*A^3*B^6*C^2+13677510*A^3*B^5*C^3+ 17255700*A^3*B^4*C^4+13677510*A^3*B^3*C^5+7381572*A^3*B^2*C^6+2692206*A^3*B*C^7 +360294*A^3*C^8+76956*A^2*B^9+947364*A^2*B^8*C+3516480*A^2*B^7*C^2+7381572*A^2* B^6*C^3+10501260*A^2*B^5*C^4+10501260*A^2*B^4*C^5+7381572*A^2*B^3*C^6+3516480*A ^2*B^2*C^7+947364*A^2*B*C^8+76956*A^2*C^9+6996*A*B^10+153912*A*B^9*C+947364*A*B ^8*C^2+2692206*A*B^7*C^3+4530240*A*B^6*C^4+5276964*A*B^5*C^5+4530240*A*B^4*C^6+ 2692206*A*B^3*C^7+947364*A*B^2*C^8+153912*A*B*C^9+6996*A*C^10+6996*B^10*C+76956 *B^9*C^2+360294*B^8*C^3+933966*B^7*C^4+1476156*B^6*C^5+1476156*B^5*C^6+933966*B ^4*C^7+360294*B^3*C^8+76956*B^2*C^9+6996*B*C^10-7425*A^8*B-7425*A^8*C-66825*A^7 *B^2-133650*A^7*B*C-66825*A^7*C^2-230175*A^6*B^3-680130*A^6*B^2*C-680130*A^6*B* C^2-230175*A^6*C^3-408375*A^5*B^4-1636470*A^5*B^3*C-2233440*A^5*B^2*C^2-1636470 *A^5*B*C^3-408375*A^5*C^4-408375*A^4*B^5-2165130*A^4*B^4*C-3767445*A^4*B^3*C^2-\ 3767445*A^4*B^2*C^3-2165130*A^4*B*C^4-408375*A^4*C^5-230175*A^3*B^6-1636470*A^3 *B^5*C-3767445*A^3*B^4*C^2-4722300*A^3*B^3*C^3-3767445*A^3*B^2*C^4-1636470*A^3* B*C^5-230175*A^3*C^6-66825*A^2*B^7-680130*A^2*B^6*C-2233440*A^2*B^5*C^2-3767445 *A^2*B^4*C^3-3767445*A^2*B^3*C^4-2233440*A^2*B^2*C^5-680130*A^2*B*C^6-66825*A^2 *C^7-7425*A*B^8-133650*A*B^7*C-680130*A*B^6*C^2-1636470*A*B^5*C^3-2165130*A*B^4 *C^4-1636470*A*B^3*C^5-680130*A*B^2*C^6-133650*A*B*C^7-7425*A*C^8-7425*B^8*C-\ 66825*B^7*C^2-230175*B^6*C^3-408375*B^5*C^4-408375*B^4*C^5-230175*B^3*C^6-66825 *B^2*C^7-7425*B*C^8-2992*A^6*B-2992*A^6*C-14960*A^5*B^2-39248*A^5*B*C-14960*A^5 *C^2-29920*A^4*B^3-130064*A^4*B^2*C-130064*A^4*B*C^2-29920*A^4*C^3-29920*A^3*B^ 4-187616*A^3*B^3*C-315392*A^3*B^2*C^2-187616*A^3*B*C^3-29920*A^3*C^4-14960*A^2* B^5-130064*A^2*B^4*C-315392*A^2*B^3*C^2-315392*A^2*B^2*C^3-130064*A^2*B*C^4-\ 14960*A^2*C^5-2992*A*B^6-39248*A*B^5*C-130064*A*B^4*C^2-187616*A*B^3*C^3-130064 *A*B^2*C^4-39248*A*B*C^5-2992*A*C^6-2992*B^6*C-14960*B^5*C^2-29920*B^4*C^3-\ 29920*B^3*C^4-14960*B^2*C^5-2992*B*C^6-1280*A^5-6400*A^4*B-6400*A^4*C-12800*A^3 *B^2-25600*A^3*B*C-12800*A^3*C^2-12800*A^2*B^3-38400*A^2*B^2*C-38400*A^2*B*C^2-\ 12800*A^2*C^3-6400*A*B^4-25600*A*B^3*C-38400*A*B^2*C^2-25600*A*B*C^3-6400*A*C^4 -1280*B^5-6400*B^4*C-12800*B^3*C^2-12800*B^2*C^3-6400*B*C^4-1280*C^5)/(A^2*B+A^ 2*C+A*B^2-3*A*B*C+A*C^2+B^2*C+B*C^2-A-B-C)^3 Note that if N is large and the starting capitals are x1*N,x2*N,x3*N, (where\ , of course, x1+x2+x3=1), then the limit of the, 6, -th scaled moment about the mean tends to 14 14 13 2 13 13 2 (10 x1 x2 + 10 x1 x3 + 150 x1 x2 + 300 x1 x2 x3 + 150 x1 x3 12 3 12 2 12 2 12 3 + 1017 x1 x2 + 2688 x1 x2 x3 + 2688 x1 x2 x3 + 1017 x1 x3 11 4 11 3 11 2 2 + 4121 x1 x2 + 12161 x1 x2 x3 + 16080 x1 x2 x3 11 3 11 4 10 5 10 4 + 12161 x1 x2 x3 + 4121 x1 x3 + 11120 x1 x2 + 33270 x1 x2 x3 10 3 2 10 2 3 10 4 + 50854 x1 x2 x3 + 50854 x1 x2 x3 + 33270 x1 x2 x3 10 5 9 6 9 5 9 4 2 + 11120 x1 x3 + 21076 x1 x2 + 61006 x1 x2 x3 + 112904 x1 x2 x3 9 3 3 9 2 4 9 5 + 145948 x1 x2 x3 + 112904 x1 x2 x3 + 61006 x1 x2 x3 9 6 8 7 8 6 8 5 2 + 21076 x1 x3 + 28842 x1 x2 + 82280 x1 x2 x3 + 211618 x1 x2 x3 8 4 3 8 3 4 8 2 5 + 340373 x1 x2 x3 + 340373 x1 x2 x3 + 211618 x1 x2 x3 8 6 8 7 7 8 7 7 + 82280 x1 x2 x3 + 28842 x1 x3 + 28842 x1 x2 + 89562 x1 x2 x3 7 6 2 7 5 3 7 4 4 + 309210 x1 x2 x3 + 406527 x1 x2 x3 + 316074 x1 x2 x3 7 3 5 7 2 6 7 7 + 406527 x1 x2 x3 + 309210 x1 x2 x3 + 89562 x1 x2 x3 7 8 6 9 6 8 6 7 2 + 28842 x1 x3 + 21076 x1 x2 + 82280 x1 x2 x3 + 309210 x1 x2 x3 6 6 3 6 5 4 6 4 5 + 344784 x1 x2 x3 + 1036860 x1 x2 x3 + 1036860 x1 x2 x3 6 3 6 6 2 7 6 8 + 344784 x1 x2 x3 + 309210 x1 x2 x3 + 82280 x1 x2 x3 6 9 5 10 5 9 5 8 2 + 21076 x1 x3 + 11120 x1 x2 + 61006 x1 x2 x3 + 211618 x1 x2 x3 5 7 3 5 6 4 5 5 5 + 406527 x1 x2 x3 + 1036860 x1 x2 x3 - 289212 x1 x2 x3 5 4 6 5 3 7 5 2 8 + 1036860 x1 x2 x3 + 406527 x1 x2 x3 + 211618 x1 x2 x3 5 9 5 10 4 11 4 10 + 61006 x1 x2 x3 + 11120 x1 x3 + 4121 x1 x2 + 33270 x1 x2 x3 4 9 2 4 8 3 4 7 4 + 112904 x1 x2 x3 + 340373 x1 x2 x3 + 316074 x1 x2 x3 4 6 5 4 5 6 4 4 7 + 1036860 x1 x2 x3 + 1036860 x1 x2 x3 + 316074 x1 x2 x3 4 3 8 4 2 9 4 10 + 340373 x1 x2 x3 + 112904 x1 x2 x3 + 33270 x1 x2 x3 4 11 3 12 3 11 3 10 2 + 4121 x1 x3 + 1017 x1 x2 + 12161 x1 x2 x3 + 50854 x1 x2 x3 3 9 3 3 8 4 3 7 5 + 145948 x1 x2 x3 + 340373 x1 x2 x3 + 406527 x1 x2 x3 3 6 6 3 5 7 3 4 8 + 344784 x1 x2 x3 + 406527 x1 x2 x3 + 340373 x1 x2 x3 3 3 9 3 2 10 3 11 + 145948 x1 x2 x3 + 50854 x1 x2 x3 + 12161 x1 x2 x3 3 12 2 13 2 12 2 11 2 + 1017 x1 x3 + 150 x1 x2 + 2688 x1 x2 x3 + 16080 x1 x2 x3 2 10 3 2 9 4 2 8 5 + 50854 x1 x2 x3 + 112904 x1 x2 x3 + 211618 x1 x2 x3 2 7 6 2 6 7 2 5 8 + 309210 x1 x2 x3 + 309210 x1 x2 x3 + 211618 x1 x2 x3 2 4 9 2 3 10 2 2 11 + 112904 x1 x2 x3 + 50854 x1 x2 x3 + 16080 x1 x2 x3 2 12 2 13 14 13 + 2688 x1 x2 x3 + 150 x1 x3 + 10 x1 x2 + 300 x1 x2 x3 12 2 11 3 10 4 + 2688 x1 x2 x3 + 12161 x1 x2 x3 + 33270 x1 x2 x3 9 5 8 6 7 7 + 61006 x1 x2 x3 + 82280 x1 x2 x3 + 89562 x1 x2 x3 6 8 5 9 4 10 + 82280 x1 x2 x3 + 61006 x1 x2 x3 + 33270 x1 x2 x3 3 11 2 12 13 14 + 12161 x1 x2 x3 + 2688 x1 x2 x3 + 300 x1 x2 x3 + 10 x1 x3 14 13 2 12 3 11 4 + 10 x2 x3 + 150 x2 x3 + 1017 x2 x3 + 4121 x2 x3 10 5 9 6 8 7 7 8 + 11120 x2 x3 + 21076 x2 x3 + 28842 x2 x3 + 28842 x2 x3 6 9 5 10 4 11 3 12 + 21076 x2 x3 + 11120 x2 x3 + 4121 x2 x3 + 1017 x2 x3 2 13 14 / 2 2 2 + 150 x2 x3 + 10 x2 x3 ) / (3080 x1 x2 x3 / 2 2 2 2 2 2 3 (x1 x2 + x1 x3 + x1 x2 - 3 x1 x2 x3 + x1 x3 + x2 x3 + x2 x3 ) ) and in Maple notation 1/3080*(10*x1^14*x2+10*x1^14*x3+150*x1^13*x2^2+300*x1^13*x2*x3+150*x1^13*x3^2+ 1017*x1^12*x2^3+2688*x1^12*x2^2*x3+2688*x1^12*x2*x3^2+1017*x1^12*x3^3+4121*x1^ 11*x2^4+12161*x1^11*x2^3*x3+16080*x1^11*x2^2*x3^2+12161*x1^11*x2*x3^3+4121*x1^ 11*x3^4+11120*x1^10*x2^5+33270*x1^10*x2^4*x3+50854*x1^10*x2^3*x3^2+50854*x1^10* x2^2*x3^3+33270*x1^10*x2*x3^4+11120*x1^10*x3^5+21076*x1^9*x2^6+61006*x1^9*x2^5* x3+112904*x1^9*x2^4*x3^2+145948*x1^9*x2^3*x3^3+112904*x1^9*x2^2*x3^4+61006*x1^9 *x2*x3^5+21076*x1^9*x3^6+28842*x1^8*x2^7+82280*x1^8*x2^6*x3+211618*x1^8*x2^5*x3 ^2+340373*x1^8*x2^4*x3^3+340373*x1^8*x2^3*x3^4+211618*x1^8*x2^2*x3^5+82280*x1^8 *x2*x3^6+28842*x1^8*x3^7+28842*x1^7*x2^8+89562*x1^7*x2^7*x3+309210*x1^7*x2^6*x3 ^2+406527*x1^7*x2^5*x3^3+316074*x1^7*x2^4*x3^4+406527*x1^7*x2^3*x3^5+309210*x1^ 7*x2^2*x3^6+89562*x1^7*x2*x3^7+28842*x1^7*x3^8+21076*x1^6*x2^9+82280*x1^6*x2^8* x3+309210*x1^6*x2^7*x3^2+344784*x1^6*x2^6*x3^3+1036860*x1^6*x2^5*x3^4+1036860* x1^6*x2^4*x3^5+344784*x1^6*x2^3*x3^6+309210*x1^6*x2^2*x3^7+82280*x1^6*x2*x3^8+ 21076*x1^6*x3^9+11120*x1^5*x2^10+61006*x1^5*x2^9*x3+211618*x1^5*x2^8*x3^2+ 406527*x1^5*x2^7*x3^3+1036860*x1^5*x2^6*x3^4-289212*x1^5*x2^5*x3^5+1036860*x1^5 *x2^4*x3^6+406527*x1^5*x2^3*x3^7+211618*x1^5*x2^2*x3^8+61006*x1^5*x2*x3^9+11120 *x1^5*x3^10+4121*x1^4*x2^11+33270*x1^4*x2^10*x3+112904*x1^4*x2^9*x3^2+340373*x1 ^4*x2^8*x3^3+316074*x1^4*x2^7*x3^4+1036860*x1^4*x2^6*x3^5+1036860*x1^4*x2^5*x3^ 6+316074*x1^4*x2^4*x3^7+340373*x1^4*x2^3*x3^8+112904*x1^4*x2^2*x3^9+33270*x1^4* x2*x3^10+4121*x1^4*x3^11+1017*x1^3*x2^12+12161*x1^3*x2^11*x3+50854*x1^3*x2^10* x3^2+145948*x1^3*x2^9*x3^3+340373*x1^3*x2^8*x3^4+406527*x1^3*x2^7*x3^5+344784* x1^3*x2^6*x3^6+406527*x1^3*x2^5*x3^7+340373*x1^3*x2^4*x3^8+145948*x1^3*x2^3*x3^ 9+50854*x1^3*x2^2*x3^10+12161*x1^3*x2*x3^11+1017*x1^3*x3^12+150*x1^2*x2^13+2688 *x1^2*x2^12*x3+16080*x1^2*x2^11*x3^2+50854*x1^2*x2^10*x3^3+112904*x1^2*x2^9*x3^ 4+211618*x1^2*x2^8*x3^5+309210*x1^2*x2^7*x3^6+309210*x1^2*x2^6*x3^7+211618*x1^2 *x2^5*x3^8+112904*x1^2*x2^4*x3^9+50854*x1^2*x2^3*x3^10+16080*x1^2*x2^2*x3^11+ 2688*x1^2*x2*x3^12+150*x1^2*x3^13+10*x1*x2^14+300*x1*x2^13*x3+2688*x1*x2^12*x3^ 2+12161*x1*x2^11*x3^3+33270*x1*x2^10*x3^4+61006*x1*x2^9*x3^5+82280*x1*x2^8*x3^6 +89562*x1*x2^7*x3^7+82280*x1*x2^6*x3^8+61006*x1*x2^5*x3^9+33270*x1*x2^4*x3^10+ 12161*x1*x2^3*x3^11+2688*x1*x2^2*x3^12+300*x1*x2*x3^13+10*x1*x3^14+10*x2^14*x3+ 150*x2^13*x3^2+1017*x2^12*x3^3+4121*x2^11*x3^4+11120*x2^10*x3^5+21076*x2^9*x3^6 +28842*x2^8*x3^7+28842*x2^7*x3^8+21076*x2^6*x3^9+11120*x2^5*x3^10+4121*x2^4*x3^ 11+1017*x2^3*x3^12+150*x2^2*x3^13+10*x2*x3^14)/x1^2/x2^2/x3^2/(x1^2*x2+x1^2*x3+ x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2*x3+x2*x3^2)^3 The , 7, -th (straight) moment of the Duration is: 11 11 10 2 10 10 2 3 A B C (6219 A B + 6219 A C + 74628 A B + 149256 A B C + 74628 A C 9 3 9 2 9 2 9 3 + 389979 A B + 1190412 A B C + 1190412 A B C + 389979 A C 8 4 8 3 8 2 2 8 3 + 1163430 A B + 4858470 A B C + 7390080 A B C + 4858470 A B C 8 4 7 5 7 4 7 3 2 + 1163430 A C + 2184057 A B + 11787606 A B C + 24469560 A B C 7 2 3 7 4 7 5 6 6 + 24469560 A B C + 11787606 A B C + 2184057 A C + 2684664 A B 6 5 6 4 2 6 3 3 + 18132552 A B C + 48584448 A B C + 66273120 A B C 6 2 4 6 5 6 6 5 7 + 48584448 A B C + 18132552 A B C + 2684664 A C + 2184057 A B 5 6 5 5 2 5 4 3 + 18132552 A B C + 60778368 A B C + 106980048 A B C 5 3 4 5 2 5 5 6 + 106980048 A B C + 60778368 A B C + 18132552 A B C 5 7 4 8 4 7 4 6 2 + 2184057 A C + 1163430 A B + 11787606 A B C + 48584448 A B C 4 5 3 4 4 4 4 3 5 + 106980048 A B C + 138039552 A B C + 106980048 A B C 4 2 6 4 7 4 8 3 9 + 48584448 A B C + 11787606 A B C + 1163430 A C + 389979 A B 3 8 3 7 2 3 6 3 + 4858470 A B C + 24469560 A B C + 66273120 A B C 3 5 4 3 4 5 3 3 6 + 106980048 A B C + 106980048 A B C + 66273120 A B C 3 2 7 3 8 3 9 2 10 + 24469560 A B C + 4858470 A B C + 389979 A C + 74628 A B 2 9 2 8 2 2 7 3 + 1190412 A B C + 7390080 A B C + 24469560 A B C 2 6 4 2 5 5 2 4 6 + 48584448 A B C + 60778368 A B C + 48584448 A B C 2 3 7 2 2 8 2 9 2 10 + 24469560 A B C + 7390080 A B C + 1190412 A B C + 74628 A C 11 10 9 2 8 3 + 6219 A B + 149256 A B C + 1190412 A B C + 4858470 A B C 7 4 6 5 5 6 + 11787606 A B C + 18132552 A B C + 18132552 A B C 4 7 3 8 2 9 10 + 11787606 A B C + 4858470 A B C + 1190412 A B C + 149256 A B C 11 11 10 2 9 3 8 4 + 6219 A C + 6219 B C + 74628 B C + 389979 B C + 1163430 B C 7 5 6 6 5 7 4 8 + 2184057 B C + 2684664 B C + 2184057 B C + 1163430 B C 3 9 2 10 11 9 9 + 389979 B C + 74628 B C + 6219 B C - 122850 A B - 122850 A C 8 2 8 8 2 7 3 - 1228500 A B - 2457000 A B C - 1228500 A C - 5122845 A B 7 2 7 2 7 3 6 4 - 15773940 A B C - 15773940 A B C - 5122845 A C - 11498760 A B 6 3 6 2 2 6 3 6 4 - 49238280 A B C - 75479040 A B C - 49238280 A B C - 11498760 A C 5 5 5 4 5 3 2 - 14987700 A B - 85208760 A B C - 181228320 A B C 5 2 3 5 4 5 5 4 6 - 181228320 A B C - 85208760 A B C - 14987700 A C - 11498760 A B 4 5 4 4 2 4 3 3 - 85208760 A B C - 240589440 A B C - 333758880 A B C 4 2 4 4 5 4 6 3 7 - 240589440 A B C - 85208760 A B C - 11498760 A C - 5122845 A B 3 6 3 5 2 3 4 3 - 49238280 A B C - 181228320 A B C - 333758880 A B C 3 3 4 3 2 5 3 6 - 333758880 A B C - 181228320 A B C - 49238280 A B C 3 7 2 8 2 7 2 6 2 - 5122845 A C - 1228500 A B - 15773940 A B C - 75479040 A B C 2 5 3 2 4 4 2 3 5 - 181228320 A B C - 240589440 A B C - 181228320 A B C 2 2 6 2 7 2 8 9 - 75479040 A B C - 15773940 A B C - 1228500 A C - 122850 A B 8 7 2 6 3 5 4 - 2457000 A B C - 15773940 A B C - 49238280 A B C - 85208760 A B C 4 5 3 6 2 7 8 - 85208760 A B C - 49238280 A B C - 15773940 A B C - 2457000 A B C 9 9 8 2 7 3 - 122850 A C - 122850 B C - 1228500 B C - 5122845 B C 6 4 5 5 4 6 3 7 - 11498760 B C - 14987700 B C - 11498760 B C - 5122845 B C 2 8 9 7 7 6 2 - 1228500 B C - 122850 B C + 765765 A B + 765765 A C + 6126120 A B 6 6 2 5 3 5 2 + 12252240 A B C + 6126120 A C + 18888870 A B + 59219160 A B C 5 2 5 3 4 4 4 3 + 59219160 A B C + 18888870 A C + 27567540 A B + 125585460 A B C 4 2 2 4 3 4 4 3 5 + 196035840 A B C + 125585460 A B C + 27567540 A C + 18888870 A B 3 4 3 3 2 3 2 3 + 125585460 A B C + 285885600 A B C + 285885600 A B C 3 4 3 5 2 6 2 5 + 125585460 A B C + 18888870 A C + 6126120 A B + 59219160 A B C 2 4 2 2 3 3 2 2 4 + 196035840 A B C + 285885600 A B C + 196035840 A B C 2 5 2 6 7 6 + 59219160 A B C + 6126120 A C + 765765 A B + 12252240 A B C 5 2 4 3 3 4 + 59219160 A B C + 125585460 A B C + 125585460 A B C 2 5 6 7 7 + 59219160 A B C + 12252240 A B C + 765765 A C + 765765 B C 6 2 5 3 4 4 3 5 + 6126120 B C + 18888870 B C + 27567540 B C + 18888870 B C 2 6 7 5 5 + 6126120 B C + 765765 B C - 1484340 A B - 1484340 A C 4 2 4 4 2 3 3 - 8906040 A B - 17812080 A B C - 8906040 A C - 17069910 A B 3 2 3 2 3 3 2 4 - 56404920 A B C - 56404920 A B C - 17069910 A C - 8906040 A B 2 3 2 2 2 2 3 2 4 - 56404920 A B C - 94997760 A B C - 56404920 A B C - 8906040 A C 5 4 3 2 2 3 - 1484340 A B - 17812080 A B C - 56404920 A B C - 56404920 A B C 4 5 5 4 2 - 17812080 A B C - 1484340 A C - 1484340 B C - 8906040 B C 3 3 2 4 5 3 - 17069910 B C - 8906040 B C - 1484340 B C + 150150 A B 3 2 2 2 2 2 3 + 150150 A C + 600600 A B + 1201200 A B C + 600600 A C + 150150 A B 2 2 3 3 + 1201200 A B C + 1201200 A B C + 150150 A C + 150150 B C 2 2 3 + 600600 B C + 150150 B C - 502320 A B - 502320 A C - 502320 B C - 361280)/(3203200 (A + B + C)) and in Maple notation 3/3203200*A*B*C*(6219*A^11*B+6219*A^11*C+74628*A^10*B^2+149256*A^10*B*C+74628*A ^10*C^2+389979*A^9*B^3+1190412*A^9*B^2*C+1190412*A^9*B*C^2+389979*A^9*C^3+ 1163430*A^8*B^4+4858470*A^8*B^3*C+7390080*A^8*B^2*C^2+4858470*A^8*B*C^3+1163430 *A^8*C^4+2184057*A^7*B^5+11787606*A^7*B^4*C+24469560*A^7*B^3*C^2+24469560*A^7*B ^2*C^3+11787606*A^7*B*C^4+2184057*A^7*C^5+2684664*A^6*B^6+18132552*A^6*B^5*C+ 48584448*A^6*B^4*C^2+66273120*A^6*B^3*C^3+48584448*A^6*B^2*C^4+18132552*A^6*B*C ^5+2684664*A^6*C^6+2184057*A^5*B^7+18132552*A^5*B^6*C+60778368*A^5*B^5*C^2+ 106980048*A^5*B^4*C^3+106980048*A^5*B^3*C^4+60778368*A^5*B^2*C^5+18132552*A^5*B *C^6+2184057*A^5*C^7+1163430*A^4*B^8+11787606*A^4*B^7*C+48584448*A^4*B^6*C^2+ 106980048*A^4*B^5*C^3+138039552*A^4*B^4*C^4+106980048*A^4*B^3*C^5+48584448*A^4* B^2*C^6+11787606*A^4*B*C^7+1163430*A^4*C^8+389979*A^3*B^9+4858470*A^3*B^8*C+ 24469560*A^3*B^7*C^2+66273120*A^3*B^6*C^3+106980048*A^3*B^5*C^4+106980048*A^3*B ^4*C^5+66273120*A^3*B^3*C^6+24469560*A^3*B^2*C^7+4858470*A^3*B*C^8+389979*A^3*C ^9+74628*A^2*B^10+1190412*A^2*B^9*C+7390080*A^2*B^8*C^2+24469560*A^2*B^7*C^3+ 48584448*A^2*B^6*C^4+60778368*A^2*B^5*C^5+48584448*A^2*B^4*C^6+24469560*A^2*B^3 *C^7+7390080*A^2*B^2*C^8+1190412*A^2*B*C^9+74628*A^2*C^10+6219*A*B^11+149256*A* B^10*C+1190412*A*B^9*C^2+4858470*A*B^8*C^3+11787606*A*B^7*C^4+18132552*A*B^6*C^ 5+18132552*A*B^5*C^6+11787606*A*B^4*C^7+4858470*A*B^3*C^8+1190412*A*B^2*C^9+ 149256*A*B*C^10+6219*A*C^11+6219*B^11*C+74628*B^10*C^2+389979*B^9*C^3+1163430*B ^8*C^4+2184057*B^7*C^5+2684664*B^6*C^6+2184057*B^5*C^7+1163430*B^4*C^8+389979*B ^3*C^9+74628*B^2*C^10+6219*B*C^11-122850*A^9*B-122850*A^9*C-1228500*A^8*B^2-\ 2457000*A^8*B*C-1228500*A^8*C^2-5122845*A^7*B^3-15773940*A^7*B^2*C-15773940*A^7 *B*C^2-5122845*A^7*C^3-11498760*A^6*B^4-49238280*A^6*B^3*C-75479040*A^6*B^2*C^2 -49238280*A^6*B*C^3-11498760*A^6*C^4-14987700*A^5*B^5-85208760*A^5*B^4*C-\ 181228320*A^5*B^3*C^2-181228320*A^5*B^2*C^3-85208760*A^5*B*C^4-14987700*A^5*C^5 -11498760*A^4*B^6-85208760*A^4*B^5*C-240589440*A^4*B^4*C^2-333758880*A^4*B^3*C^ 3-240589440*A^4*B^2*C^4-85208760*A^4*B*C^5-11498760*A^4*C^6-5122845*A^3*B^7-\ 49238280*A^3*B^6*C-181228320*A^3*B^5*C^2-333758880*A^3*B^4*C^3-333758880*A^3*B^ 3*C^4-181228320*A^3*B^2*C^5-49238280*A^3*B*C^6-5122845*A^3*C^7-1228500*A^2*B^8-\ 15773940*A^2*B^7*C-75479040*A^2*B^6*C^2-181228320*A^2*B^5*C^3-240589440*A^2*B^4 *C^4-181228320*A^2*B^3*C^5-75479040*A^2*B^2*C^6-15773940*A^2*B*C^7-1228500*A^2* C^8-122850*A*B^9-2457000*A*B^8*C-15773940*A*B^7*C^2-49238280*A*B^6*C^3-85208760 *A*B^5*C^4-85208760*A*B^4*C^5-49238280*A*B^3*C^6-15773940*A*B^2*C^7-2457000*A*B *C^8-122850*A*C^9-122850*B^9*C-1228500*B^8*C^2-5122845*B^7*C^3-11498760*B^6*C^4 -14987700*B^5*C^5-11498760*B^4*C^6-5122845*B^3*C^7-1228500*B^2*C^8-122850*B*C^9 +765765*A^7*B+765765*A^7*C+6126120*A^6*B^2+12252240*A^6*B*C+6126120*A^6*C^2+ 18888870*A^5*B^3+59219160*A^5*B^2*C+59219160*A^5*B*C^2+18888870*A^5*C^3+ 27567540*A^4*B^4+125585460*A^4*B^3*C+196035840*A^4*B^2*C^2+125585460*A^4*B*C^3+ 27567540*A^4*C^4+18888870*A^3*B^5+125585460*A^3*B^4*C+285885600*A^3*B^3*C^2+ 285885600*A^3*B^2*C^3+125585460*A^3*B*C^4+18888870*A^3*C^5+6126120*A^2*B^6+ 59219160*A^2*B^5*C+196035840*A^2*B^4*C^2+285885600*A^2*B^3*C^3+196035840*A^2*B^ 2*C^4+59219160*A^2*B*C^5+6126120*A^2*C^6+765765*A*B^7+12252240*A*B^6*C+59219160 *A*B^5*C^2+125585460*A*B^4*C^3+125585460*A*B^3*C^4+59219160*A*B^2*C^5+12252240* A*B*C^6+765765*A*C^7+765765*B^7*C+6126120*B^6*C^2+18888870*B^5*C^3+27567540*B^4 *C^4+18888870*B^3*C^5+6126120*B^2*C^6+765765*B*C^7-1484340*A^5*B-1484340*A^5*C-\ 8906040*A^4*B^2-17812080*A^4*B*C-8906040*A^4*C^2-17069910*A^3*B^3-56404920*A^3* B^2*C-56404920*A^3*B*C^2-17069910*A^3*C^3-8906040*A^2*B^4-56404920*A^2*B^3*C-\ 94997760*A^2*B^2*C^2-56404920*A^2*B*C^3-8906040*A^2*C^4-1484340*A*B^5-17812080* A*B^4*C-56404920*A*B^3*C^2-56404920*A*B^2*C^3-17812080*A*B*C^4-1484340*A*C^5-\ 1484340*B^5*C-8906040*B^4*C^2-17069910*B^3*C^3-8906040*B^2*C^4-1484340*B*C^5+ 150150*A^3*B+150150*A^3*C+600600*A^2*B^2+1201200*A^2*B*C+600600*A^2*C^2+150150* A*B^3+1201200*A*B^2*C+1201200*A*B*C^2+150150*A*C^3+150150*B^3*C+600600*B^2*C^2+ 150150*B*C^3-502320*A*B-502320*A*C-502320*B*C-361280)/(A+B+C) Hence the, 7, -th moment about the mean is 17 17 16 2 16 3 A B C (6219 A B + 6219 A C + 111942 A B + 223884 A B C 16 2 15 3 15 2 15 2 + 111942 A C + 931032 A B + 2567871 A B C + 2567871 A B C 15 3 14 4 14 3 14 2 2 + 931032 A C + 4747104 A B + 15630516 A B C + 21766824 A B C 14 3 14 4 13 5 13 4 + 15630516 A B C + 4747104 A C + 16600167 A B + 60416091 A B C 13 3 2 13 2 3 13 4 + 102328515 A B C + 102328515 A B C + 60416091 A B C 13 5 12 6 12 5 + 16600167 A C + 42196770 A B + 162636894 A B C 12 4 2 12 3 3 12 2 4 + 318992814 A B C + 397105380 A B C + 318992814 A B C 12 5 12 6 11 7 + 162636894 A B C + 42196770 A C + 80625168 A B 11 6 11 5 2 11 4 3 + 323561277 A B C + 735232680 A B C + 1102557456 A B C 11 3 4 11 2 5 11 6 + 1102557456 A B C + 735232680 A B C + 323561277 A B C 11 7 10 8 10 7 + 80625168 A C + 118084824 A B + 497295396 A B C 10 6 2 10 5 3 10 4 4 + 1331140824 A B C + 2304980496 A B C + 2706100488 A B C 10 3 5 10 2 6 10 7 + 2304980496 A B C + 1331140824 A B C + 497295396 A B C 10 8 9 9 9 8 + 118084824 A C + 133956108 A B + 610599132 A B C 9 7 2 9 6 3 9 5 4 + 1918085598 A B C + 3581413407 A B C + 4698355662 A B C 9 4 5 9 3 6 9 2 7 + 4698355662 A B C + 3581413407 A B C + 1918085598 A B C 9 8 9 9 8 10 + 610599132 A B C + 133956108 A C + 118084824 A B 8 9 8 8 2 8 7 3 + 610599132 A B C + 2172404520 A B C + 4299673950 A B C 8 6 4 8 5 5 8 4 6 + 7136229672 A B C + 9032891868 A B C + 7136229672 A B C 8 3 7 8 2 8 8 9 + 4299673950 A B C + 2172404520 A B C + 610599132 A B C 8 10 7 11 7 10 + 118084824 A C + 80625168 A B + 497295396 A B C 7 9 2 7 8 3 7 7 4 + 1918085598 A B C + 4299673950 A B C + 8609481738 A B C 7 6 5 7 5 6 7 4 7 + 8970064389 A B C + 8970064389 A B C + 8609481738 A B C 7 3 8 7 2 9 7 10 + 4299673950 A B C + 1918085598 A B C + 497295396 A B C 7 11 6 12 6 11 + 80625168 A C + 42196770 A B + 323561277 A B C 6 10 2 6 9 3 6 8 4 + 1331140824 A B C + 3581413407 A B C + 7136229672 A B C 6 7 5 6 6 6 6 5 7 + 8970064389 A B C + 15836833584 A B C + 8970064389 A B C 6 4 8 6 3 9 6 2 10 + 7136229672 A B C + 3581413407 A B C + 1331140824 A B C 6 11 6 12 5 13 + 323561277 A B C + 42196770 A C + 16600167 A B 5 12 5 11 2 5 10 3 + 162636894 A B C + 735232680 A B C + 2304980496 A B C 5 9 4 5 8 5 5 7 6 + 4698355662 A B C + 9032891868 A B C + 8970064389 A B C 5 6 7 5 5 8 5 4 9 + 8970064389 A B C + 9032891868 A B C + 4698355662 A B C 5 3 10 5 2 11 5 12 + 2304980496 A B C + 735232680 A B C + 162636894 A B C 5 13 4 14 4 13 + 16600167 A C + 4747104 A B + 60416091 A B C 4 12 2 4 11 3 4 10 4 + 318992814 A B C + 1102557456 A B C + 2706100488 A B C 4 9 5 4 8 6 4 7 7 + 4698355662 A B C + 7136229672 A B C + 8609481738 A B C 4 6 8 4 5 9 4 4 10 + 7136229672 A B C + 4698355662 A B C + 2706100488 A B C 4 3 11 4 2 12 4 13 + 1102557456 A B C + 318992814 A B C + 60416091 A B C 4 14 3 15 3 14 3 13 2 + 4747104 A C + 931032 A B + 15630516 A B C + 102328515 A B C 3 12 3 3 11 4 3 10 5 + 397105380 A B C + 1102557456 A B C + 2304980496 A B C 3 9 6 3 8 7 3 7 8 + 3581413407 A B C + 4299673950 A B C + 4299673950 A B C 3 6 9 3 5 10 3 4 11 + 3581413407 A B C + 2304980496 A B C + 1102557456 A B C 3 3 12 3 2 13 3 14 + 397105380 A B C + 102328515 A B C + 15630516 A B C 3 15 2 16 2 15 2 14 2 + 931032 A C + 111942 A B + 2567871 A B C + 21766824 A B C 2 13 3 2 12 4 2 11 5 + 102328515 A B C + 318992814 A B C + 735232680 A B C 2 10 6 2 9 7 2 8 8 + 1331140824 A B C + 1918085598 A B C + 2172404520 A B C 2 7 9 2 6 10 2 5 11 + 1918085598 A B C + 1331140824 A B C + 735232680 A B C 2 4 12 2 3 13 2 2 14 + 318992814 A B C + 102328515 A B C + 21766824 A B C 2 15 2 16 17 16 + 2567871 A B C + 111942 A C + 6219 A B + 223884 A B C 15 2 14 3 13 4 + 2567871 A B C + 15630516 A B C + 60416091 A B C 12 5 11 6 10 7 + 162636894 A B C + 323561277 A B C + 497295396 A B C 9 8 8 9 7 10 + 610599132 A B C + 610599132 A B C + 497295396 A B C 6 11 5 12 4 13 + 323561277 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3841330350 A B C - 1386656700 A B C 2 9 2 10 11 10 - 263238690 A B C - 17812080 A C - 1484340 A B - 35624160 A B C 9 2 8 3 7 4 - 263238690 A B C - 967931250 A B C - 2141400690 A B C 6 5 5 6 4 7 - 3122926950 A B C - 3122926950 A B C - 2141400690 A B C 3 8 2 9 10 11 - 967931250 A B C - 263238690 A B C - 35624160 A B C - 1484340 A C 11 10 2 9 3 8 4 - 1484340 B C - 17812080 B C - 92771250 B C - 274602900 B C 7 5 6 6 5 7 4 8 - 511355130 B C - 626391480 B C - 511355130 B C - 274602900 B C 3 9 2 10 11 9 - 92771250 B C - 17812080 B C - 1484340 B C + 150150 A B 9 8 2 8 8 2 + 150150 A C + 1501500 A B + 3003000 A B C + 1501500 A C 7 3 7 2 7 2 7 3 + 6006000 A B + 26936910 A B C + 26936910 A B C + 6006000 A C 6 4 6 3 6 2 2 + 12912900 A B + 97147050 A B C + 185585400 A B C 6 3 6 4 5 5 5 4 + 97147050 A B C + 12912900 A C + 16516500 A B + 176426250 A B C 5 3 2 5 2 3 5 4 + 523933410 A B C + 523933410 A B C + 176426250 A B C 5 5 4 6 4 5 4 4 2 + 16516500 A C + 12912900 A B + 176426250 A B C + 727566840 A 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10046400 A B - 12776400 A B C - 5460000 A B C 4 3 4 4 3 5 3 4 - 12776400 A B C - 10046400 A C - 7534800 A B - 12776400 A B C 3 3 2 3 2 3 3 4 3 5 - 655200 A B C - 655200 A B C - 12776400 A B C - 7534800 A C 2 6 2 5 2 4 2 2 3 3 - 3013920 A B - 8146320 A B C - 5460000 A B C - 655200 A B C 2 2 4 2 5 2 6 7 - 5460000 A B C - 8146320 A B C - 3013920 A C - 502320 A B 6 5 2 4 3 3 4 - 3035760 A B C - 8146320 A B C - 12776400 A B C - 12776400 A B C 2 5 6 7 7 - 8146320 A B C - 3035760 A B C - 502320 A C - 502320 B C 6 2 5 3 4 4 3 5 - 3013920 B C - 7534800 B C - 10046400 B C - 7534800 B C 2 6 7 6 5 5 - 3013920 B C - 502320 B C - 361280 A - 2167680 A B - 2167680 A C 4 2 4 4 2 3 3 - 5419200 A B - 10838400 A B C - 5419200 A C - 7225600 A B 3 2 3 2 3 3 2 4 - 21676800 A B C - 21676800 A B C - 7225600 A C - 5419200 A B 2 3 2 2 2 2 3 2 4 - 21676800 A B C - 32515200 A B C - 21676800 A B C - 5419200 A C 5 4 3 2 2 3 - 2167680 A B - 10838400 A B C - 21676800 A B C - 21676800 A B C 4 5 6 5 - 10838400 A B C - 2167680 A C - 361280 B - 2167680 B C 4 2 3 3 2 4 5 6 - 5419200 B C - 7225600 B C - 5419200 B C - 2167680 B C - 361280 C / 7 ) / (3203200 (A + B + C) ) / and in Maple notation 3/3203200*A*B*C*(6219*A^17*B+6219*A^17*C+111942*A^16*B^2+223884*A^16*B*C+111942 *A^16*C^2+931032*A^15*B^3+2567871*A^15*B^2*C+2567871*A^15*B*C^2+931032*A^15*C^3 +4747104*A^14*B^4+15630516*A^14*B^3*C+21766824*A^14*B^2*C^2+15630516*A^14*B*C^3 +4747104*A^14*C^4+16600167*A^13*B^5+60416091*A^13*B^4*C+102328515*A^13*B^3*C^2+ 102328515*A^13*B^2*C^3+60416091*A^13*B*C^4+16600167*A^13*C^5+42196770*A^12*B^6+ 162636894*A^12*B^5*C+318992814*A^12*B^4*C^2+397105380*A^12*B^3*C^3+318992814*A^ 12*B^2*C^4+162636894*A^12*B*C^5+42196770*A^12*C^6+80625168*A^11*B^7+323561277*A ^11*B^6*C+735232680*A^11*B^5*C^2+1102557456*A^11*B^4*C^3+1102557456*A^11*B^3*C^ 4+735232680*A^11*B^2*C^5+323561277*A^11*B*C^6+80625168*A^11*C^7+118084824*A^10* B^8+497295396*A^10*B^7*C+1331140824*A^10*B^6*C^2+2304980496*A^10*B^5*C^3+ 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601636035*B^9*C^5+1007236230*B^8*C^6+1193061870*B^7*C^7+1007236230*B^6*C^8+ 601636035*B^5*C^9+248107860*B^4*C^10+67132065*B^3*C^11+10720710*B^2*C^12+765765 *B*C^13-1484340*A^11*B-1484340*A^11*C-17812080*A^10*B^2-35624160*A^10*B*C-\ 17812080*A^10*C^2-92771250*A^9*B^3-263238690*A^9*B^2*C-263238690*A^9*B*C^2-\ 92771250*A^9*C^3-274602900*A^8*B^4-967931250*A^8*B^3*C-1386656700*A^8*B^2*C^2-\ 967931250*A^8*B*C^3-274602900*A^8*C^4-511355130*A^7*B^5-2141400690*A^7*B^4*C-\ 3841330350*A^7*B^3*C^2-3841330350*A^7*B^2*C^3-2141400690*A^7*B*C^4-511355130*A^ 7*C^5-626391480*A^6*B^6-3122926950*A^6*B^5*C-6651928140*A^6*B^4*C^2-8510308950* A^6*B^3*C^3-6651928140*A^6*B^2*C^4-3122926950*A^6*B*C^5-626391480*A^6*C^6-\ 511355130*A^5*B^7-3122926950*A^5*B^6*C-7903655760*A^5*B^5*C^2-12264852600*A^5*B ^4*C^3-12264852600*A^5*B^3*C^4-7903655760*A^5*B^2*C^5-3122926950*A^5*B*C^6-\ 511355130*A^5*C^7-274602900*A^4*B^8-2141400690*A^4*B^7*C-6651928140*A^4*B^6*C^2 -12264852600*A^4*B^5*C^3-14959444500*A^4*B^4*C^4-12264852600*A^4*B^3*C^5-\ 6651928140*A^4*B^2*C^6-2141400690*A^4*B*C^7-274602900*A^4*C^8-92771250*A^3*B^9-\ 967931250*A^3*B^8*C-3841330350*A^3*B^7*C^2-8510308950*A^3*B^6*C^3-12264852600*A ^3*B^5*C^4-12264852600*A^3*B^4*C^5-8510308950*A^3*B^3*C^6-3841330350*A^3*B^2*C^ 7-967931250*A^3*B*C^8-92771250*A^3*C^9-17812080*A^2*B^10-263238690*A^2*B^9*C-\ 1386656700*A^2*B^8*C^2-3841330350*A^2*B^7*C^3-6651928140*A^2*B^6*C^4-7903655760 *A^2*B^5*C^5-6651928140*A^2*B^4*C^6-3841330350*A^2*B^3*C^7-1386656700*A^2*B^2*C ^8-263238690*A^2*B*C^9-17812080*A^2*C^10-1484340*A*B^11-35624160*A*B^10*C-\ 263238690*A*B^9*C^2-967931250*A*B^8*C^3-2141400690*A*B^7*C^4-3122926950*A*B^6*C ^5-3122926950*A*B^5*C^6-2141400690*A*B^4*C^7-967931250*A*B^3*C^8-263238690*A*B^ 2*C^9-35624160*A*B*C^10-1484340*A*C^11-1484340*B^11*C-17812080*B^10*C^2-\ 92771250*B^9*C^3-274602900*B^8*C^4-511355130*B^7*C^5-626391480*B^6*C^6-\ 511355130*B^5*C^7-274602900*B^4*C^8-92771250*B^3*C^9-17812080*B^2*C^10-1484340* B*C^11+150150*A^9*B+150150*A^9*C+1501500*A^8*B^2+3003000*A^8*B*C+1501500*A^8*C^ 2+6006000*A^7*B^3+26936910*A^7*B^2*C+26936910*A^7*B*C^2+6006000*A^7*C^3+ 12912900*A^6*B^4+97147050*A^6*B^3*C+185585400*A^6*B^2*C^2+97147050*A^6*B*C^3+ 12912900*A^6*C^4+16516500*A^5*B^5+176426250*A^5*B^4*C+523933410*A^5*B^3*C^2+ 523933410*A^5*B^2*C^3+176426250*A^5*B*C^4+16516500*A^5*C^5+12912900*A^4*B^6+ 176426250*A^4*B^5*C+727566840*A^4*B^4*C^2+1128106980*A^4*B^3*C^3+727566840*A^4* B^2*C^4+176426250*A^4*B*C^5+12912900*A^4*C^6+6006000*A^3*B^7+97147050*A^3*B^6*C +523933410*A^3*B^5*C^2+1128106980*A^3*B^4*C^3+1128106980*A^3*B^3*C^4+523933410* A^3*B^2*C^5+97147050*A^3*B*C^6+6006000*A^3*C^7+1501500*A^2*B^8+26936910*A^2*B^7 *C+185585400*A^2*B^6*C^2+523933410*A^2*B^5*C^3+727566840*A^2*B^4*C^4+523933410* A^2*B^3*C^5+185585400*A^2*B^2*C^6+26936910*A^2*B*C^7+1501500*A^2*C^8+150150*A*B ^9+3003000*A*B^8*C+26936910*A*B^7*C^2+97147050*A*B^6*C^3+176426250*A*B^5*C^4+ 176426250*A*B^4*C^5+97147050*A*B^3*C^6+26936910*A*B^2*C^7+3003000*A*B*C^8+ 150150*A*C^9+150150*B^9*C+1501500*B^8*C^2+6006000*B^7*C^3+12912900*B^6*C^4+ 16516500*B^5*C^5+12912900*B^4*C^6+6006000*B^3*C^7+1501500*B^2*C^8+150150*B*C^9-\ 502320*A^7*B-502320*A^7*C-3013920*A^6*B^2-3035760*A^6*B*C-3013920*A^6*C^2-\ 7534800*A^5*B^3-8146320*A^5*B^2*C-8146320*A^5*B*C^2-7534800*A^5*C^3-10046400*A^ 4*B^4-12776400*A^4*B^3*C-5460000*A^4*B^2*C^2-12776400*A^4*B*C^3-10046400*A^4*C^ 4-7534800*A^3*B^5-12776400*A^3*B^4*C-655200*A^3*B^3*C^2-655200*A^3*B^2*C^3-\ 12776400*A^3*B*C^4-7534800*A^3*C^5-3013920*A^2*B^6-8146320*A^2*B^5*C-5460000*A^ 2*B^4*C^2-655200*A^2*B^3*C^3-5460000*A^2*B^2*C^4-8146320*A^2*B*C^5-3013920*A^2* C^6-502320*A*B^7-3035760*A*B^6*C-8146320*A*B^5*C^2-12776400*A*B^4*C^3-12776400* A*B^3*C^4-8146320*A*B^2*C^5-3035760*A*B*C^6-502320*A*C^7-502320*B^7*C-3013920*B ^6*C^2-7534800*B^5*C^3-10046400*B^4*C^4-7534800*B^3*C^5-3013920*B^2*C^6-502320* B*C^7-361280*A^6-2167680*A^5*B-2167680*A^5*C-5419200*A^4*B^2-10838400*A^4*B*C-\ 5419200*A^4*C^2-7225600*A^3*B^3-21676800*A^3*B^2*C-21676800*A^3*B*C^2-7225600*A ^3*C^3-5419200*A^2*B^4-21676800*A^2*B^3*C-32515200*A^2*B^2*C^2-21676800*A^2*B*C ^3-5419200*A^2*C^4-2167680*A*B^5-10838400*A*B^4*C-21676800*A*B^3*C^2-21676800*A *B^2*C^3-10838400*A*B*C^4-2167680*A*C^5-361280*B^6-2167680*B^5*C-5419200*B^4*C^ 2-7225600*B^3*C^3-5419200*B^2*C^4-2167680*B*C^5-361280*C^6)/(A+B+C)^7 Hence the scaled, 7, -th moment about the mean is 17 17 16 2 16 16 2 A B C (6219 A B + 6219 A C + 111942 A B + 223884 A B C + 111942 A C 15 3 15 2 15 2 15 3 + 931032 A B + 2567871 A B C + 2567871 A B C + 931032 A C 14 4 14 3 14 2 2 + 4747104 A B + 15630516 A B C + 21766824 A B C 14 3 14 4 13 5 13 4 + 15630516 A B C + 4747104 A C + 16600167 A B + 60416091 A B C 13 3 2 13 2 3 13 4 + 102328515 A B C + 102328515 A B C + 60416091 A B C 13 5 12 6 12 5 + 16600167 A C + 42196770 A B + 162636894 A B C 12 4 2 12 3 3 12 2 4 + 318992814 A B C + 397105380 A B C + 318992814 A B C 12 5 12 6 11 7 + 162636894 A B C + 42196770 A C + 80625168 A B 11 6 11 5 2 11 4 3 + 323561277 A B C + 735232680 A B C + 1102557456 A B C 11 3 4 11 2 5 11 6 + 1102557456 A B C + 735232680 A B C + 323561277 A B C 11 7 10 8 10 7 + 80625168 A C + 118084824 A B + 497295396 A B C 10 6 2 10 5 3 10 4 4 + 1331140824 A B C + 2304980496 A B C + 2706100488 A B C 10 3 5 10 2 6 10 7 + 2304980496 A B C + 1331140824 A B C + 497295396 A B C 10 8 9 9 9 8 + 118084824 A C + 133956108 A B + 610599132 A B C 9 7 2 9 6 3 9 5 4 + 1918085598 A B C + 3581413407 A B C + 4698355662 A B C 9 4 5 9 3 6 9 2 7 + 4698355662 A B C + 3581413407 A B C + 1918085598 A B C 9 8 9 9 8 10 + 610599132 A B C + 133956108 A C + 118084824 A B 8 9 8 8 2 8 7 3 + 610599132 A B C + 2172404520 A B C + 4299673950 A B C 8 6 4 8 5 5 8 4 6 + 7136229672 A B C + 9032891868 A B C + 7136229672 A B C 8 3 7 8 2 8 8 9 + 4299673950 A B C + 2172404520 A B C + 610599132 A B C 8 10 7 11 7 10 + 118084824 A C + 80625168 A B + 497295396 A B C 7 9 2 7 8 3 7 7 4 + 1918085598 A B C + 4299673950 A B C + 8609481738 A B C 7 6 5 7 5 6 7 4 7 + 8970064389 A B C + 8970064389 A B C + 8609481738 A B C 7 3 8 7 2 9 7 10 + 4299673950 A B C + 1918085598 A B C + 497295396 A B C 7 11 6 12 6 11 + 80625168 A C + 42196770 A B + 323561277 A B C 6 10 2 6 9 3 6 8 4 + 1331140824 A B C + 3581413407 A B C + 7136229672 A B C 6 7 5 6 6 6 6 5 7 + 8970064389 A B C + 15836833584 A B C + 8970064389 A B C 6 4 8 6 3 9 6 2 10 + 7136229672 A B C + 3581413407 A B C + 1331140824 A B C 6 11 6 12 5 13 + 323561277 A B C + 42196770 A C + 16600167 A B 5 12 5 11 2 5 10 3 + 162636894 A B C + 735232680 A B C + 2304980496 A B C 5 9 4 5 8 5 5 7 6 + 4698355662 A B C + 9032891868 A B C + 8970064389 A B C 5 6 7 5 5 8 5 4 9 + 8970064389 A B C + 9032891868 A B C + 4698355662 A B C 5 3 10 5 2 11 5 12 + 2304980496 A B C + 735232680 A B C + 162636894 A B C 5 13 4 14 4 13 + 16600167 A C + 4747104 A B + 60416091 A B C 4 12 2 4 11 3 4 10 4 + 318992814 A B C + 1102557456 A B C + 2706100488 A B C 4 9 5 4 8 6 4 7 7 + 4698355662 A B C + 7136229672 A B C + 8609481738 A B C 4 6 8 4 5 9 4 4 10 + 7136229672 A B C + 4698355662 A B C + 2706100488 A B C 4 3 11 4 2 12 4 13 + 1102557456 A B C + 318992814 A B C + 60416091 A B C 4 14 3 15 3 14 3 13 2 + 4747104 A C + 931032 A B + 15630516 A B C + 102328515 A B C 3 12 3 3 11 4 3 10 5 + 397105380 A B C + 1102557456 A B C + 2304980496 A B C 3 9 6 3 8 7 3 7 8 + 3581413407 A B C + 4299673950 A B C + 4299673950 A B C 3 6 9 3 5 10 3 4 11 + 3581413407 A B C + 2304980496 A B C + 1102557456 A B C 3 3 12 3 2 13 3 14 + 397105380 A B C + 102328515 A B C + 15630516 A B C 3 15 2 16 2 15 2 14 2 + 931032 A C + 111942 A B + 2567871 A B C + 21766824 A B C 2 13 3 2 12 4 2 11 5 + 102328515 A B C + 318992814 A B C + 735232680 A B C 2 10 6 2 9 7 2 8 8 + 1331140824 A B C + 1918085598 A B C + 2172404520 A B C 2 7 9 2 6 10 2 5 11 + 1918085598 A B C + 1331140824 A B C + 735232680 A B C 2 4 12 2 3 13 2 2 14 + 318992814 A B C + 102328515 A B C + 21766824 A B C 2 15 2 16 17 16 + 2567871 A B C + 111942 A C + 6219 A B + 223884 A B C 15 2 14 3 13 4 + 2567871 A B C + 15630516 A B C + 60416091 A B C 12 5 11 6 10 7 + 162636894 A B C + 323561277 A B C + 497295396 A B C 9 8 8 9 7 10 + 610599132 A B C + 610599132 A B C + 497295396 A B C 6 11 5 12 4 13 + 323561277 A B C + 162636894 A B C + 60416091 A B C 3 14 2 15 16 17 + 15630516 A B C + 2567871 A B C + 223884 A B C + 6219 A C 17 16 2 15 3 14 4 + 6219 B C + 111942 B C + 931032 B C + 4747104 B C 13 5 12 6 11 7 10 8 + 16600167 B C + 42196770 B C + 80625168 B C + 118084824 B C 9 9 8 10 7 11 6 12 + 133956108 B C + 118084824 B C + 80625168 B C + 42196770 B C 5 13 4 14 3 15 2 16 + 16600167 B C + 4747104 B C + 931032 B C + 111942 B C 17 15 15 14 2 + 6219 B C - 122850 A B - 122850 A C - 1965600 A B 14 14 2 13 3 13 2 - 3931200 A B C - 1965600 A C - 14336595 A B - 39361140 A B C 13 2 13 3 12 4 - 39361140 A B C - 14336595 A C - 63120330 A B 12 3 12 2 2 12 3 - 205454340 A B C - 284668020 A B C - 205454340 A B C 12 4 11 5 11 4 - 63120330 A C - 187235685 A B - 669286800 A B C 11 3 2 11 2 3 11 4 - 1123573815 A B C - 1123573815 A B C - 669286800 A B C 11 5 10 6 10 5 - 187235685 A C - 395527860 A B - 1496951820 A B C 10 4 2 10 3 3 10 2 4 - 2934542520 A B C - 3666237120 A B C - 2934542520 A B C 10 5 10 6 9 7 - 1496951820 A B C - 395527860 A C - 613242630 A B 9 6 9 5 2 9 4 3 - 2454051600 A B C - 5705669970 A B C - 8666883225 A B C 9 3 4 9 2 5 9 6 - 8666883225 A B C - 5705669970 A B C - 2454051600 A B C 9 7 8 8 8 7 - 613242630 A C - 708647940 A B - 3097023930 A B C 8 6 2 8 5 3 8 4 4 - 8592964380 A B C - 14719444740 A B C - 17029712700 A B C 8 3 5 8 2 6 8 7 - 14719444740 A B C - 8592964380 A B C - 3097023930 A B C 8 8 7 9 7 8 - 708647940 A C - 613242630 A B - 3097023930 A B C 7 7 2 7 6 3 7 5 4 - 9891071190 A B C - 18449035605 A B C - 25891190325 A B C 7 4 5 7 3 6 7 2 7 - 25891190325 A B C - 18449035605 A B C - 9891071190 A B C 7 8 7 9 6 10 - 3097023930 A B C - 613242630 A C - 395527860 A B 6 9 6 8 2 6 7 3 - 2454051600 A B C - 8592964380 A B C - 18449035605 A B C 6 6 4 6 5 5 6 4 6 - 30399969600 A B C - 34341046740 A B C - 30399969600 A B C 6 3 7 6 2 8 6 9 - 18449035605 A B C - 8592964380 A B C - 2454051600 A B C 6 10 5 11 5 10 - 395527860 A C - 187235685 A B - 1496951820 A B C 5 9 2 5 8 3 5 7 4 - 5705669970 A B C - 14719444740 A B C - 25891190325 A B C 5 6 5 5 5 6 5 4 7 - 34341046740 A B C - 34341046740 A B C - 25891190325 A B C 5 3 8 5 2 9 5 10 - 14719444740 A B C - 5705669970 A B C - 1496951820 A B C 5 11 4 12 4 11 - 187235685 A C - 63120330 A B - 669286800 A B C 4 10 2 4 9 3 4 8 4 - 2934542520 A B C - 8666883225 A B C - 17029712700 A B C 4 7 5 4 6 6 4 5 7 - 25891190325 A B C - 30399969600 A B C - 25891190325 A B C 4 4 8 4 3 9 4 2 10 - 17029712700 A B C - 8666883225 A B C - 2934542520 A B C 4 11 4 12 3 13 - 669286800 A B C - 63120330 A C - 14336595 A B 3 12 3 11 2 3 10 3 - 205454340 A B C - 1123573815 A B C - 3666237120 A B C 3 9 4 3 8 5 3 7 6 - 8666883225 A B C - 14719444740 A B C - 18449035605 A B C 3 6 7 3 5 8 3 4 9 - 18449035605 A B C - 14719444740 A B C - 8666883225 A B C 3 3 10 3 2 11 3 12 - 3666237120 A B C - 1123573815 A B C - 205454340 A B C 3 13 2 14 2 13 - 14336595 A C - 1965600 A B - 39361140 A B C 2 12 2 2 11 3 2 10 4 - 284668020 A B C - 1123573815 A B C - 2934542520 A B C 2 9 5 2 8 6 2 7 7 - 5705669970 A B C - 8592964380 A B C - 9891071190 A B C 2 6 8 2 5 9 2 4 10 - 8592964380 A B C - 5705669970 A B C - 2934542520 A B C 2 3 11 2 2 12 2 13 - 1123573815 A B C - 284668020 A B C - 39361140 A B C 2 14 15 14 13 2 - 1965600 A C - 122850 A B - 3931200 A B C - 39361140 A B C 12 3 11 4 10 5 - 205454340 A B C - 669286800 A B C - 1496951820 A B C 9 6 8 7 7 8 - 2454051600 A B C - 3097023930 A B C - 3097023930 A B C 6 9 5 10 4 11 - 2454051600 A B C - 1496951820 A B C - 669286800 A B C 3 12 2 13 14 15 - 205454340 A B C - 39361140 A B C - 3931200 A B C - 122850 A C 15 14 2 13 3 12 4 - 122850 B C - 1965600 B C - 14336595 B C - 63120330 B C 11 5 10 6 9 7 8 8 - 187235685 B C - 395527860 B C - 613242630 B C - 708647940 B C 7 9 6 10 5 11 4 12 - 613242630 B C - 395527860 B C - 187235685 B C - 63120330 B C 3 13 2 14 15 13 - 14336595 B C - 1965600 B C - 122850 B C + 765765 A B 13 12 2 12 12 2 + 765765 A C + 10720710 A B + 21441420 A B C + 10720710 A C 11 3 11 2 11 2 + 67132065 A B + 184849665 A B C + 184849665 A B C 11 3 10 4 10 3 + 67132065 A C + 248107860 A B + 812972160 A B C 10 2 2 10 3 10 4 + 1129728600 A B C + 812972160 A B C + 248107860 A C 9 5 9 4 9 3 2 + 601636035 A B + 2185898715 A B C + 3701918220 A B C 9 2 3 9 4 9 5 + 3701918220 A B C + 2185898715 A B C + 601636035 A C 8 6 8 5 8 4 2 + 1007236230 A B + 3983779800 A B C + 8029511490 A B C 8 3 3 8 2 4 8 5 + 10105935840 A B C + 8029511490 A B C + 3983779800 A B C 8 6 7 7 7 6 + 1007236230 A C + 1193061870 A B + 5274904635 A B C 7 5 2 7 4 3 7 3 4 + 12641143515 A B C + 19108314225 A B C + 19108314225 A B C 7 2 5 7 6 7 7 + 12641143515 A B C + 5274904635 A B C + 1193061870 A C 6 8 6 7 6 6 2 + 1007236230 A B + 5274904635 A B C + 14715901200 A B C 6 5 3 6 4 4 6 3 5 + 26130394290 A B C + 30935104200 A B C + 26130394290 A B C 6 2 6 6 7 6 8 + 14715901200 A B C + 5274904635 A B C + 1007236230 A C 5 9 5 8 5 7 2 + 601636035 A B + 3983779800 A B C + 12641143515 A B C 5 6 3 5 5 4 5 4 5 + 26130394290 A B C + 35837021220 A B C + 35837021220 A B C 5 3 6 5 2 7 5 8 + 26130394290 A B C + 12641143515 A B C + 3983779800 A B C 5 9 4 10 4 9 + 601636035 A C + 248107860 A B + 2185898715 A B C 4 8 2 4 7 3 4 6 4 + 8029511490 A B C + 19108314225 A B C + 30935104200 A B C 4 5 5 4 4 6 4 3 7 + 35837021220 A B C + 30935104200 A B C + 19108314225 A B C 4 2 8 4 9 4 10 + 8029511490 A B C + 2185898715 A B C + 248107860 A C 3 11 3 10 3 9 2 + 67132065 A B + 812972160 A B C + 3701918220 A B C 3 8 3 3 7 4 3 6 5 + 10105935840 A B C + 19108314225 A B C + 26130394290 A B C 3 5 6 3 4 7 3 3 8 + 26130394290 A B C + 19108314225 A B C + 10105935840 A B C 3 2 9 3 10 3 11 + 3701918220 A B C + 812972160 A B C + 67132065 A C 2 12 2 11 2 10 2 + 10720710 A B + 184849665 A B C + 1129728600 A B C 2 9 3 2 8 4 2 7 5 + 3701918220 A B C + 8029511490 A B C + 12641143515 A B C 2 6 6 2 5 7 2 4 8 + 14715901200 A B C + 12641143515 A B C + 8029511490 A B C 2 3 9 2 2 10 2 11 + 3701918220 A B C + 1129728600 A B C + 184849665 A B C 2 12 13 12 11 2 + 10720710 A C + 765765 A B + 21441420 A B C + 184849665 A B C 10 3 9 4 8 5 + 812972160 A B C + 2185898715 A B C + 3983779800 A B C 7 6 6 7 5 8 + 5274904635 A B C + 5274904635 A B C + 3983779800 A B C 4 9 3 10 2 11 + 2185898715 A B C + 812972160 A B C + 184849665 A B C 12 13 13 12 2 + 21441420 A B C + 765765 A C + 765765 B C + 10720710 B C 11 3 10 4 9 5 8 6 + 67132065 B C + 248107860 B C + 601636035 B C + 1007236230 B C 7 7 6 8 5 9 4 10 + 1193061870 B C + 1007236230 B C + 601636035 B C + 248107860 B C 3 11 2 12 13 11 + 67132065 B C + 10720710 B C + 765765 B C - 1484340 A B 11 10 2 10 10 2 - 1484340 A C - 17812080 A B - 35624160 A B C - 17812080 A C 9 3 9 2 9 2 9 3 - 92771250 A B - 263238690 A B C - 263238690 A B C - 92771250 A C 8 4 8 3 8 2 2 - 274602900 A B - 967931250 A B C - 1386656700 A B C 8 3 8 4 7 5 - 967931250 A B C - 274602900 A C - 511355130 A B 7 4 7 3 2 7 2 3 - 2141400690 A B C - 3841330350 A B C - 3841330350 A B C 7 4 7 5 6 6 - 2141400690 A B C - 511355130 A C - 626391480 A B 6 5 6 4 2 6 3 3 - 3122926950 A B C - 6651928140 A B C - 8510308950 A B C 6 2 4 6 5 6 6 - 6651928140 A B C - 3122926950 A B C - 626391480 A C 5 7 5 6 5 5 2 - 511355130 A B - 3122926950 A B C - 7903655760 A B C 5 4 3 5 3 4 5 2 5 - 12264852600 A B C - 12264852600 A B C - 7903655760 A B C 5 6 5 7 4 8 - 3122926950 A B C - 511355130 A C - 274602900 A B 4 7 4 6 2 4 5 3 - 2141400690 A B C - 6651928140 A B C - 12264852600 A B C 4 4 4 4 3 5 4 2 6 - 14959444500 A B C - 12264852600 A B C - 6651928140 A B C 4 7 4 8 3 9 - 2141400690 A B C - 274602900 A C - 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2934542520*A^4*B^2*C^10-669286800*A^4*B*C^11-63120330*A^4*C^12-14336595*A^3*B^ 13-205454340*A^3*B^12*C-1123573815*A^3*B^11*C^2-3666237120*A^3*B^10*C^3-\ 8666883225*A^3*B^9*C^4-14719444740*A^3*B^8*C^5-18449035605*A^3*B^7*C^6-\ 18449035605*A^3*B^6*C^7-14719444740*A^3*B^5*C^8-8666883225*A^3*B^4*C^9-\ 3666237120*A^3*B^3*C^10-1123573815*A^3*B^2*C^11-205454340*A^3*B*C^12-14336595*A ^3*C^13-1965600*A^2*B^14-39361140*A^2*B^13*C-284668020*A^2*B^12*C^2-1123573815* A^2*B^11*C^3-2934542520*A^2*B^10*C^4-5705669970*A^2*B^9*C^5-8592964380*A^2*B^8* C^6-9891071190*A^2*B^7*C^7-8592964380*A^2*B^6*C^8-5705669970*A^2*B^5*C^9-\ 2934542520*A^2*B^4*C^10-1123573815*A^2*B^3*C^11-284668020*A^2*B^2*C^12-39361140 *A^2*B*C^13-1965600*A^2*C^14-122850*A*B^15-3931200*A*B^14*C-39361140*A*B^13*C^2 -205454340*A*B^12*C^3-669286800*A*B^11*C^4-1496951820*A*B^10*C^5-2454051600*A*B ^9*C^6-3097023930*A*B^8*C^7-3097023930*A*B^7*C^8-2454051600*A*B^6*C^9-\ 1496951820*A*B^5*C^10-669286800*A*B^4*C^11-205454340*A*B^3*C^12-39361140*A*B^2* C^13-3931200*A*B*C^14-122850*A*C^15-122850*B^15*C-1965600*B^14*C^2-14336595*B^ 13*C^3-63120330*B^12*C^4-187235685*B^11*C^5-395527860*B^10*C^6-613242630*B^9*C^ 7-708647940*B^8*C^8-613242630*B^7*C^9-395527860*B^6*C^10-187235685*B^5*C^11-\ 63120330*B^4*C^12-14336595*B^3*C^13-1965600*B^2*C^14-122850*B*C^15+765765*A^13* B+765765*A^13*C+10720710*A^12*B^2+21441420*A^12*B*C+10720710*A^12*C^2+67132065* A^11*B^3+184849665*A^11*B^2*C+184849665*A^11*B*C^2+67132065*A^11*C^3+248107860* A^10*B^4+812972160*A^10*B^3*C+1129728600*A^10*B^2*C^2+812972160*A^10*B*C^3+ 248107860*A^10*C^4+601636035*A^9*B^5+2185898715*A^9*B^4*C+3701918220*A^9*B^3*C^ 2+3701918220*A^9*B^2*C^3+2185898715*A^9*B*C^4+601636035*A^9*C^5+1007236230*A^8* B^6+3983779800*A^8*B^5*C+8029511490*A^8*B^4*C^2+10105935840*A^8*B^3*C^3+ 8029511490*A^8*B^2*C^4+3983779800*A^8*B*C^5+1007236230*A^8*C^6+1193061870*A^7*B ^7+5274904635*A^7*B^6*C+12641143515*A^7*B^5*C^2+19108314225*A^7*B^4*C^3+ 19108314225*A^7*B^3*C^4+12641143515*A^7*B^2*C^5+5274904635*A^7*B*C^6+1193061870 *A^7*C^7+1007236230*A^6*B^8+5274904635*A^6*B^7*C+14715901200*A^6*B^6*C^2+ 26130394290*A^6*B^5*C^3+30935104200*A^6*B^4*C^4+26130394290*A^6*B^3*C^5+ 14715901200*A^6*B^2*C^6+5274904635*A^6*B*C^7+1007236230*A^6*C^8+601636035*A^5*B ^9+3983779800*A^5*B^8*C+12641143515*A^5*B^7*C^2+26130394290*A^5*B^6*C^3+ 35837021220*A^5*B^5*C^4+35837021220*A^5*B^4*C^5+26130394290*A^5*B^3*C^6+ 12641143515*A^5*B^2*C^7+3983779800*A^5*B*C^8+601636035*A^5*C^9+248107860*A^4*B^ 10+2185898715*A^4*B^9*C+8029511490*A^4*B^8*C^2+19108314225*A^4*B^7*C^3+ 30935104200*A^4*B^6*C^4+35837021220*A^4*B^5*C^5+30935104200*A^4*B^4*C^6+ 19108314225*A^4*B^3*C^7+8029511490*A^4*B^2*C^8+2185898715*A^4*B*C^9+248107860*A ^4*C^10+67132065*A^3*B^11+812972160*A^3*B^10*C+3701918220*A^3*B^9*C^2+ 10105935840*A^3*B^8*C^3+19108314225*A^3*B^7*C^4+26130394290*A^3*B^6*C^5+ 26130394290*A^3*B^5*C^6+19108314225*A^3*B^4*C^7+10105935840*A^3*B^3*C^8+ 3701918220*A^3*B^2*C^9+812972160*A^3*B*C^10+67132065*A^3*C^11+10720710*A^2*B^12 +184849665*A^2*B^11*C+1129728600*A^2*B^10*C^2+3701918220*A^2*B^9*C^3+8029511490 *A^2*B^8*C^4+12641143515*A^2*B^7*C^5+14715901200*A^2*B^6*C^6+12641143515*A^2*B^ 5*C^7+8029511490*A^2*B^4*C^8+3701918220*A^2*B^3*C^9+1129728600*A^2*B^2*C^10+ 184849665*A^2*B*C^11+10720710*A^2*C^12+765765*A*B^13+21441420*A*B^12*C+ 184849665*A*B^11*C^2+812972160*A*B^10*C^3+2185898715*A*B^9*C^4+3983779800*A*B^8 *C^5+5274904635*A*B^7*C^6+5274904635*A*B^6*C^7+3983779800*A*B^5*C^8+2185898715* A*B^4*C^9+812972160*A*B^3*C^10+184849665*A*B^2*C^11+21441420*A*B*C^12+765765*A* C^13+765765*B^13*C+10720710*B^12*C^2+67132065*B^11*C^3+248107860*B^10*C^4+ 601636035*B^9*C^5+1007236230*B^8*C^6+1193061870*B^7*C^7+1007236230*B^6*C^8+ 601636035*B^5*C^9+248107860*B^4*C^10+67132065*B^3*C^11+10720710*B^2*C^12+765765 *B*C^13-1484340*A^11*B-1484340*A^11*C-17812080*A^10*B^2-35624160*A^10*B*C-\ 17812080*A^10*C^2-92771250*A^9*B^3-263238690*A^9*B^2*C-263238690*A^9*B*C^2-\ 92771250*A^9*C^3-274602900*A^8*B^4-967931250*A^8*B^3*C-1386656700*A^8*B^2*C^2-\ 967931250*A^8*B*C^3-274602900*A^8*C^4-511355130*A^7*B^5-2141400690*A^7*B^4*C-\ 3841330350*A^7*B^3*C^2-3841330350*A^7*B^2*C^3-2141400690*A^7*B*C^4-511355130*A^ 7*C^5-626391480*A^6*B^6-3122926950*A^6*B^5*C-6651928140*A^6*B^4*C^2-8510308950* A^6*B^3*C^3-6651928140*A^6*B^2*C^4-3122926950*A^6*B*C^5-626391480*A^6*C^6-\ 511355130*A^5*B^7-3122926950*A^5*B^6*C-7903655760*A^5*B^5*C^2-12264852600*A^5*B ^4*C^3-12264852600*A^5*B^3*C^4-7903655760*A^5*B^2*C^5-3122926950*A^5*B*C^6-\ 511355130*A^5*C^7-274602900*A^4*B^8-2141400690*A^4*B^7*C-6651928140*A^4*B^6*C^2 -12264852600*A^4*B^5*C^3-14959444500*A^4*B^4*C^4-12264852600*A^4*B^3*C^5-\ 6651928140*A^4*B^2*C^6-2141400690*A^4*B*C^7-274602900*A^4*C^8-92771250*A^3*B^9-\ 967931250*A^3*B^8*C-3841330350*A^3*B^7*C^2-8510308950*A^3*B^6*C^3-12264852600*A ^3*B^5*C^4-12264852600*A^3*B^4*C^5-8510308950*A^3*B^3*C^6-3841330350*A^3*B^2*C^ 7-967931250*A^3*B*C^8-92771250*A^3*C^9-17812080*A^2*B^10-263238690*A^2*B^9*C-\ 1386656700*A^2*B^8*C^2-3841330350*A^2*B^7*C^3-6651928140*A^2*B^6*C^4-7903655760 *A^2*B^5*C^5-6651928140*A^2*B^4*C^6-3841330350*A^2*B^3*C^7-1386656700*A^2*B^2*C ^8-263238690*A^2*B*C^9-17812080*A^2*C^10-1484340*A*B^11-35624160*A*B^10*C-\ 263238690*A*B^9*C^2-967931250*A*B^8*C^3-2141400690*A*B^7*C^4-3122926950*A*B^6*C ^5-3122926950*A*B^5*C^6-2141400690*A*B^4*C^7-967931250*A*B^3*C^8-263238690*A*B^ 2*C^9-35624160*A*B*C^10-1484340*A*C^11-1484340*B^11*C-17812080*B^10*C^2-\ 92771250*B^9*C^3-274602900*B^8*C^4-511355130*B^7*C^5-626391480*B^6*C^6-\ 511355130*B^5*C^7-274602900*B^4*C^8-92771250*B^3*C^9-17812080*B^2*C^10-1484340* B*C^11+150150*A^9*B+150150*A^9*C+1501500*A^8*B^2+3003000*A^8*B*C+1501500*A^8*C^ 2+6006000*A^7*B^3+26936910*A^7*B^2*C+26936910*A^7*B*C^2+6006000*A^7*C^3+ 12912900*A^6*B^4+97147050*A^6*B^3*C+185585400*A^6*B^2*C^2+97147050*A^6*B*C^3+ 12912900*A^6*C^4+16516500*A^5*B^5+176426250*A^5*B^4*C+523933410*A^5*B^3*C^2+ 523933410*A^5*B^2*C^3+176426250*A^5*B*C^4+16516500*A^5*C^5+12912900*A^4*B^6+ 176426250*A^4*B^5*C+727566840*A^4*B^4*C^2+1128106980*A^4*B^3*C^3+727566840*A^4* B^2*C^4+176426250*A^4*B*C^5+12912900*A^4*C^6+6006000*A^3*B^7+97147050*A^3*B^6*C +523933410*A^3*B^5*C^2+1128106980*A^3*B^4*C^3+1128106980*A^3*B^3*C^4+523933410* A^3*B^2*C^5+97147050*A^3*B*C^6+6006000*A^3*C^7+1501500*A^2*B^8+26936910*A^2*B^7 *C+185585400*A^2*B^6*C^2+523933410*A^2*B^5*C^3+727566840*A^2*B^4*C^4+523933410* A^2*B^3*C^5+185585400*A^2*B^2*C^6+26936910*A^2*B*C^7+1501500*A^2*C^8+150150*A*B ^9+3003000*A*B^8*C+26936910*A*B^7*C^2+97147050*A*B^6*C^3+176426250*A*B^5*C^4+ 176426250*A*B^4*C^5+97147050*A*B^3*C^6+26936910*A*B^2*C^7+3003000*A*B*C^8+ 150150*A*C^9+150150*B^9*C+1501500*B^8*C^2+6006000*B^7*C^3+12912900*B^6*C^4+ 16516500*B^5*C^5+12912900*B^4*C^6+6006000*B^3*C^7+1501500*B^2*C^8+150150*B*C^9-\ 502320*A^7*B-502320*A^7*C-3013920*A^6*B^2-3035760*A^6*B*C-3013920*A^6*C^2-\ 7534800*A^5*B^3-8146320*A^5*B^2*C-8146320*A^5*B*C^2-7534800*A^5*C^3-10046400*A^ 4*B^4-12776400*A^4*B^3*C-5460000*A^4*B^2*C^2-12776400*A^4*B*C^3-10046400*A^4*C^ 4-7534800*A^3*B^5-12776400*A^3*B^4*C-655200*A^3*B^3*C^2-655200*A^3*B^2*C^3-\ 12776400*A^3*B*C^4-7534800*A^3*C^5-3013920*A^2*B^6-8146320*A^2*B^5*C-5460000*A^ 2*B^4*C^2-655200*A^2*B^3*C^3-5460000*A^2*B^2*C^4-8146320*A^2*B*C^5-3013920*A^2* C^6-502320*A*B^7-3035760*A*B^6*C-8146320*A*B^5*C^2-12776400*A*B^4*C^3-12776400* A*B^3*C^4-8146320*A*B^2*C^5-3035760*A*B*C^6-502320*A*C^7-502320*B^7*C-3013920*B ^6*C^2-7534800*B^5*C^3-10046400*B^4*C^4-7534800*B^3*C^5-3013920*B^2*C^6-502320* B*C^7-361280*A^6-2167680*A^5*B-2167680*A^5*C-5419200*A^4*B^2-10838400*A^4*B*C-\ 5419200*A^4*C^2-7225600*A^3*B^3-21676800*A^3*B^2*C-21676800*A^3*B*C^2-7225600*A ^3*C^3-5419200*A^2*B^4-21676800*A^2*B^3*C-32515200*A^2*B^2*C^2-21676800*A^2*B*C ^3-5419200*A^2*C^4-2167680*A*B^5-10838400*A*B^4*C-21676800*A*B^3*C^2-21676800*A *B^2*C^3-10838400*A*B*C^4-2167680*A*C^5-361280*B^6-2167680*B^5*C-5419200*B^4*C^ 2-7225600*B^3*C^3-5419200*B^2*C^4-2167680*B*C^5-361280*C^6)/(A+B+C)^7*3^(1/2)*2 ^(1/2)/(A*B*C*(A^2*B+A^2*C+A*B^2-3*A*B*C+A*C^2+B^2*C+B*C^2-A-B-C)/(A+B+C)^2)^(7 /2) Note that if N is large and the starting capitals are x1*N,x2*N,x3*N, (where\ , of course, x1+x2+x3=1), then the limit of the, 7, -th scaled moment about the mean tends to 3 16 2 1/2 1/2 3 15 3 1/2 1/2 (285319 x1 x2 x3 3 2 + 2418536 x1 x2 x3 3 2 3 14 4 1/2 1/2 3 13 5 1/2 1/2 + 11369835 x1 x2 x3 3 2 + 35443646 x1 x2 x3 3 2 3 12 6 1/2 1/2 3 11 7 1/2 1/2 + 81692520 x1 x2 x3 3 2 + 147904536 x1 x2 x3 3 2 3 10 8 1/2 1/2 3 9 9 1/2 1/2 + 213120622 x1 x2 x3 3 2 + 241378280 x1 x2 x3 3 2 3 8 10 1/2 1/2 3 7 11 1/2 1/2 + 213120622 x1 x2 x3 3 2 + 147904536 x1 x2 x3 3 2 3 6 12 1/2 1/2 3 5 13 1/2 1/2 + 81692520 x1 x2 x3 3 2 + 35443646 x1 x2 x3 3 2 3 4 14 1/2 1/2 3 3 15 1/2 1/2 + 11369835 x1 x2 x3 3 2 + 2418536 x1 x2 x3 3 2 3 2 16 1/2 1/2 3 17 1/2 1/2 + 285319 x1 x2 x3 3 2 + 12438 x1 x2 x3 3 2 2 18 1/2 1/2 2 17 2 1/2 1/2 + 691 x1 x2 x3 3 2 + 24876 x1 x2 x3 3 2 2 16 3 1/2 1/2 2 15 4 1/2 1/2 + 285319 x1 x2 x3 3 2 + 1736724 x1 x2 x3 3 2 2 14 5 1/2 1/2 2 13 6 1/2 1/2 + 6712899 x1 x2 x3 3 2 + 18070766 x1 x2 x3 3 2 2 12 7 1/2 1/2 2 11 8 1/2 1/2 + 35951253 x1 x2 x3 3 2 + 55255044 x1 x2 x3 3 2 2 10 9 1/2 1/2 2 9 10 1/2 1/2 + 67844348 x1 x2 x3 3 2 + 67844348 x1 x2 x3 3 2 2 8 11 1/2 1/2 2 7 12 1/2 1/2 + 55255044 x1 x2 x3 3 2 + 35951253 x1 x2 x3 3 2 2 6 13 1/2 1/2 2 5 14 1/2 1/2 + 18070766 x1 x2 x3 3 2 + 6712899 x1 x2 x3 3 2 2 4 15 1/2 1/2 2 3 16 1/2 1/2 + 1736724 x1 x2 x3 3 2 + 285319 x1 x2 x3 3 2 2 2 17 1/2 1/2 2 18 1/2 1/2 + 24876 x1 x2 x3 3 2 + 691 x1 x2 x3 3 2 18 2 1/2 1/2 17 3 1/2 1/2 + 691 x1 x2 x3 3 2 + 12438 x1 x2 x3 3 2 16 4 1/2 1/2 15 5 1/2 1/2 + 103448 x1 x2 x3 3 2 + 527456 x1 x2 x3 3 2 14 6 1/2 1/2 13 7 1/2 1/2 + 1844463 x1 x2 x3 3 2 + 4688530 x1 x2 x3 3 2 12 8 1/2 1/2 11 9 1/2 1/2 + 8958352 x1 x2 x3 3 2 + 13120536 x1 x2 x3 3 2 10 10 1/2 1/2 9 11 1/2 1/2 + 14884012 x1 x2 x3 3 2 + 13120536 x1 x2 x3 3 2 8 12 1/2 1/2 7 13 1/2 1/2 + 8958352 x1 x2 x3 3 2 + 4688530 x1 x2 x3 3 2 6 14 1/2 1/2 5 15 1/2 1/2 + 1844463 x1 x2 x3 3 2 + 527456 x1 x2 x3 3 2 4 16 1/2 1/2 3 17 1/2 1/2 + 103448 x1 x2 x3 3 2 + 12438 x1 x2 x3 3 2 2 18 1/2 1/2 11 5 5 1/2 1/2 + 691 x1 x2 x3 3 2 + 300677832 x1 x2 x3 3 2 11 4 6 1/2 1/2 11 3 7 1/2 1/2 + 256108944 x1 x2 x3 3 2 + 147904536 x1 x2 x3 3 2 11 2 8 1/2 1/2 11 9 1/2 1/2 + 55255044 x1 x2 x3 3 2 + 13120536 x1 x2 x3 3 2 10 10 1/2 1/2 10 9 2 1/2 1/2 + 14884012 x1 x2 x3 3 2 + 67844348 x1 x2 x3 3 2 10 8 3 1/2 1/2 10 7 4 1/2 1/2 + 213120622 x1 x2 x3 3 2 + 397934823 x1 x2 x3 3 2 10 6 5 1/2 1/2 10 5 6 1/2 1/2 + 522039518 x1 x2 x3 3 2 + 522039518 x1 x2 x3 3 2 10 4 7 1/2 1/2 10 3 8 1/2 1/2 + 397934823 x1 x2 x3 3 2 + 213120622 x1 x2 x3 3 2 10 2 9 1/2 1/2 10 10 1/2 1/2 + 67844348 x1 x2 x3 3 2 + 14884012 x1 x2 x3 3 2 9 11 1/2 1/2 9 10 2 1/2 1/2 + 13120536 x1 x2 x3 3 2 + 67844348 x1 x2 x3 3 2 9 9 3 1/2 1/2 9 8 4 1/2 1/2 + 241378280 x1 x2 x3 3 2 + 477741550 x1 x2 x3 3 2 9 7 5 1/2 1/2 9 6 6 1/2 1/2 + 792914408 x1 x2 x3 3 2 + 1003654652 x1 x2 x3 3 2 9 5 7 1/2 1/2 9 4 8 1/2 1/2 + 792914408 x1 x2 x3 3 2 + 477741550 x1 x2 x3 3 2 9 3 9 1/2 1/2 18 2 1/2 1/2 + 241378280 x1 x2 x3 3 2 + 691 x1 x2 x3 3 2 18 2 1/2 1/2 17 3 1/2 1/2 + 691 x1 x2 x3 3 2 + 12438 x1 x2 x3 3 2 17 2 2 1/2 1/2 17 3 1/2 1/2 + 24876 x1 x2 x3 3 2 + 12438 x1 x2 x3 3 2 16 4 1/2 1/2 16 3 2 1/2 1/2 + 103448 x1 x2 x3 3 2 + 285319 x1 x2 x3 3 2 16 2 3 1/2 1/2 16 4 1/2 1/2 + 285319 x1 x2 x3 3 2 + 103448 x1 x2 x3 3 2 15 5 1/2 1/2 5 15 1/2 1/2 + 527456 x1 x2 x3 3 2 + 527456 x1 x2 x3 3 2 5 14 2 1/2 1/2 5 13 3 1/2 1/2 + 6712899 x1 x2 x3 3 2 + 35443646 x1 x2 x3 3 2 5 12 4 1/2 1/2 5 11 5 1/2 1/2 + 122506384 x1 x2 x3 3 2 + 300677832 x1 x2 x3 3 2 5 10 6 1/2 1/2 5 9 7 1/2 1/2 + 522039518 x1 x2 x3 3 2 + 792914408 x1 x2 x3 3 2 5 8 8 1/2 1/2 5 7 9 1/2 1/2 + 956609082 x1 x2 x3 3 2 + 792914408 x1 x2 x3 3 2 5 6 10 1/2 1/2 5 5 11 1/2 1/2 + 522039518 x1 x2 x3 3 2 + 300677832 x1 x2 x3 3 2 5 4 12 1/2 1/2 5 3 13 1/2 1/2 + 122506384 x1 x2 x3 3 2 + 35443646 x1 x2 x3 3 2 5 2 14 1/2 1/2 5 15 1/2 1/2 + 6712899 x1 x2 x3 3 2 + 527456 x1 x2 x3 3 2 4 16 1/2 1/2 4 15 2 1/2 1/2 + 103448 x1 x2 x3 3 2 + 1736724 x1 x2 x3 3 2 4 14 3 1/2 1/2 4 13 4 1/2 1/2 + 11369835 x1 x2 x3 3 2 + 44122820 x1 x2 x3 3 2 4 12 5 1/2 1/2 4 11 6 1/2 1/2 + 122506384 x1 x2 x3 3 2 + 256108944 x1 x2 x3 3 2 4 10 7 1/2 1/2 4 9 8 1/2 1/2 + 397934823 x1 x2 x3 3 2 + 477741550 x1 x2 x3 3 2 4 8 9 1/2 1/2 4 7 10 1/2 1/2 + 477741550 x1 x2 x3 3 2 + 397934823 x1 x2 x3 3 2 4 6 11 1/2 1/2 4 5 12 1/2 1/2 + 256108944 x1 x2 x3 3 2 + 122506384 x1 x2 x3 3 2 4 4 13 1/2 1/2 4 3 14 1/2 1/2 + 44122820 x1 x2 x3 3 2 + 11369835 x1 x2 x3 3 2 4 2 15 1/2 1/2 4 16 1/2 1/2 + 1736724 x1 x2 x3 3 2 + 103448 x1 x2 x3 3 2 3 17 1/2 1/2 9 2 10 1/2 1/2 + 12438 x1 x2 x3 3 2 + 67844348 x1 x2 x3 3 2 9 11 1/2 1/2 8 12 1/2 1/2 + 13120536 x1 x2 x3 3 2 + 8958352 x1 x2 x3 3 2 8 11 2 1/2 1/2 8 10 3 1/2 1/2 + 55255044 x1 x2 x3 3 2 + 213120622 x1 x2 x3 3 2 8 9 4 1/2 1/2 8 8 5 1/2 1/2 + 477741550 x1 x2 x3 3 2 + 956609082 x1 x2 x3 3 2 8 7 6 1/2 1/2 8 6 7 1/2 1/2 + 996673821 x1 x2 x3 3 2 + 996673821 x1 x2 x3 3 2 8 5 8 1/2 1/2 8 4 9 1/2 1/2 + 956609082 x1 x2 x3 3 2 + 477741550 x1 x2 x3 3 2 8 3 10 1/2 1/2 8 2 11 1/2 1/2 + 213120622 x1 x2 x3 3 2 + 55255044 x1 x2 x3 3 2 8 12 1/2 1/2 7 13 1/2 1/2 + 8958352 x1 x2 x3 3 2 + 4688530 x1 x2 x3 3 2 7 12 2 1/2 1/2 7 11 3 1/2 1/2 + 35951253 x1 x2 x3 3 2 + 147904536 x1 x2 x3 3 2 7 10 4 1/2 1/2 7 9 5 1/2 1/2 + 397934823 x1 x2 x3 3 2 + 792914408 x1 x2 x3 3 2 7 8 6 1/2 1/2 7 7 7 1/2 1/2 + 996673821 x1 x2 x3 3 2 + 1759648176 x1 x2 x3 3 2 7 6 8 1/2 1/2 7 5 9 1/2 1/2 + 996673821 x1 x2 x3 3 2 + 792914408 x1 x2 x3 3 2 7 4 10 1/2 1/2 7 3 11 1/2 1/2 + 397934823 x1 x2 x3 3 2 + 147904536 x1 x2 x3 3 2 7 2 12 1/2 1/2 7 13 1/2 1/2 + 35951253 x1 x2 x3 3 2 + 4688530 x1 x2 x3 3 2 6 14 1/2 1/2 6 13 2 1/2 1/2 + 1844463 x1 x2 x3 3 2 + 18070766 x1 x2 x3 3 2 6 12 3 1/2 1/2 6 11 4 1/2 1/2 + 81692520 x1 x2 x3 3 2 + 256108944 x1 x2 x3 3 2 6 10 5 1/2 1/2 6 9 6 1/2 1/2 + 522039518 x1 x2 x3 3 2 + 1003654652 x1 x2 x3 3 2 6 8 7 1/2 1/2 6 7 8 1/2 1/2 + 996673821 x1 x2 x3 3 2 + 996673821 x1 x2 x3 3 2 6 6 9 1/2 1/2 6 5 10 1/2 1/2 + 1003654652 x1 x2 x3 3 2 + 522039518 x1 x2 x3 3 2 6 4 11 1/2 1/2 6 3 12 1/2 1/2 + 256108944 x1 x2 x3 3 2 + 81692520 x1 x2 x3 3 2 6 2 13 1/2 1/2 6 14 1/2 1/2 + 18070766 x1 x2 x3 3 2 + 1844463 x1 x2 x3 3 2 15 4 2 1/2 1/2 15 3 3 1/2 1/2 + 1736724 x1 x2 x3 3 2 + 2418536 x1 x2 x3 3 2 15 2 4 1/2 1/2 15 5 1/2 1/2 + 1736724 x1 x2 x3 3 2 + 527456 x1 x2 x3 3 2 14 6 1/2 1/2 14 5 2 1/2 1/2 + 1844463 x1 x2 x3 3 2 + 6712899 x1 x2 x3 3 2 14 4 3 1/2 1/2 14 3 4 1/2 1/2 + 11369835 x1 x2 x3 3 2 + 11369835 x1 x2 x3 3 2 14 2 5 1/2 1/2 14 6 1/2 1/2 + 6712899 x1 x2 x3 3 2 + 1844463 x1 x2 x3 3 2 13 7 1/2 1/2 13 6 2 1/2 1/2 + 4688530 x1 x2 x3 3 2 + 18070766 x1 x2 x3 3 2 13 5 3 1/2 1/2 13 4 4 1/2 1/2 + 35443646 x1 x2 x3 3 2 + 44122820 x1 x2 x3 3 2 13 3 5 1/2 1/2 13 2 6 1/2 1/2 + 35443646 x1 x2 x3 3 2 + 18070766 x1 x2 x3 3 2 13 7 1/2 1/2 12 8 1/2 1/2 + 4688530 x1 x2 x3 3 2 + 8958352 x1 x2 x3 3 2 12 7 2 1/2 1/2 12 6 3 1/2 1/2 + 35951253 x1 x2 x3 3 2 + 81692520 x1 x2 x3 3 2 12 5 4 1/2 1/2 12 4 5 1/2 1/2 + 122506384 x1 x2 x3 3 2 + 122506384 x1 x2 x3 3 2 12 3 6 1/2 1/2 12 2 7 1/2 1/2 + 81692520 x1 x2 x3 3 2 + 35951253 x1 x2 x3 3 2 12 8 1/2 1/2 11 9 1/2 1/2 + 8958352 x1 x2 x3 3 2 + 13120536 x1 x2 x3 3 2 11 8 2 1/2 1/2 11 7 3 1/2 1/2 + 55255044 x1 x2 x3 3 2 + 147904536 x1 x2 x3 3 2 11 6 4 1/2 1/2 / 3 3 + 256108944 x1 x2 x3 3 2 ) / (168168000 %1 x1 x2 x3 / 3 2 2 3 3 + 252252000 %1 x1 x2 x3 + 168168000 %1 x1 x2 x3 2 4 2 3 2 + 126126000 %1 x1 x2 x3 + 252252000 %1 x1 x2 x3 2 2 3 2 4 + 252252000 %1 x1 x2 x3 + 126126000 %1 x1 x2 x3 5 4 2 + 50450400 %1 x1 x2 x3 + 126126000 %1 x1 x2 x3 3 3 2 4 + 168168000 %1 x1 x2 x3 + 126126000 %1 x1 x2 x3 5 5 4 2 + 50450400 %1 x1 x2 x3 + 50450400 %1 x1 x2 x3 + 126126000 %1 x1 x2 x3 4 2 6 6 + 126126000 %1 x1 x2 x3 + 8408400 %1 x1 x2 + 8408400 %1 x1 x3 5 2 5 2 4 3 + 25225200 %1 x1 x2 + 25225200 %1 x1 x3 + 42042000 %1 x1 x2 4 3 3 4 3 4 + 42042000 %1 x1 x3 + 42042000 %1 x1 x2 + 42042000 %1 x1 x3 2 5 2 5 6 + 25225200 %1 x1 x2 + 25225200 %1 x1 x3 + 8408400 %1 x1 x2 6 6 5 2 + 8408400 %1 x1 x3 + 8408400 %1 x2 x3 + 25225200 %1 x2 x3 4 3 3 4 2 5 + 42042000 %1 x2 x3 + 42042000 %1 x2 x3 + 25225200 %1 x2 x3 6 7 7 7 + 8408400 %1 x2 x3 + 1201200 %1 x1 + 1201200 %1 x2 + 1201200 %1 x3 ) / %1 := |x1 x2 x3 \ 2 2 2 2 2 2 / (x1 x2 + x1 x3 + x1 x2 - 3 x1 x2 x3 + x1 x3 + x2 x3 + x2 x3 ) / ( / 2 2 2 \(7/2) x1 + 2 x1 x2 + 2 x1 x3 + x2 + 2 x2 x3 + x3 )| / and in Maple notation (285319*x1^3*x2^16*x3^2*3^(1/2)*2^(1/2)+2418536*x1^3*x2^15*x3^3*3^(1/2)*2^(1/2) +11369835*x1^3*x2^14*x3^4*3^(1/2)*2^(1/2)+35443646*x1^3*x2^13*x3^5*3^(1/2)*2^(1 /2)+81692520*x1^3*x2^12*x3^6*3^(1/2)*2^(1/2)+147904536*x1^3*x2^11*x3^7*3^(1/2)* 2^(1/2)+213120622*x1^3*x2^10*x3^8*3^(1/2)*2^(1/2)+241378280*x1^3*x2^9*x3^9*3^(1 /2)*2^(1/2)+213120622*x1^3*x2^8*x3^10*3^(1/2)*2^(1/2)+147904536*x1^3*x2^7*x3^11 *3^(1/2)*2^(1/2)+81692520*x1^3*x2^6*x3^12*3^(1/2)*2^(1/2)+35443646*x1^3*x2^5*x3 ^13*3^(1/2)*2^(1/2)+11369835*x1^3*x2^4*x3^14*3^(1/2)*2^(1/2)+2418536*x1^3*x2^3* x3^15*3^(1/2)*2^(1/2)+285319*x1^3*x2^2*x3^16*3^(1/2)*2^(1/2)+12438*x1^3*x2*x3^ 17*3^(1/2)*2^(1/2)+691*x1^2*x2^18*x3*3^(1/2)*2^(1/2)+24876*x1^2*x2^17*x3^2*3^(1 /2)*2^(1/2)+285319*x1^2*x2^16*x3^3*3^(1/2)*2^(1/2)+1736724*x1^2*x2^15*x3^4*3^(1 /2)*2^(1/2)+6712899*x1^2*x2^14*x3^5*3^(1/2)*2^(1/2)+18070766*x1^2*x2^13*x3^6*3^ (1/2)*2^(1/2)+35951253*x1^2*x2^12*x3^7*3^(1/2)*2^(1/2)+55255044*x1^2*x2^11*x3^8 *3^(1/2)*2^(1/2)+67844348*x1^2*x2^10*x3^9*3^(1/2)*2^(1/2)+67844348*x1^2*x2^9*x3 ^10*3^(1/2)*2^(1/2)+55255044*x1^2*x2^8*x3^11*3^(1/2)*2^(1/2)+35951253*x1^2*x2^7 *x3^12*3^(1/2)*2^(1/2)+18070766*x1^2*x2^6*x3^13*3^(1/2)*2^(1/2)+6712899*x1^2*x2 ^5*x3^14*3^(1/2)*2^(1/2)+1736724*x1^2*x2^4*x3^15*3^(1/2)*2^(1/2)+285319*x1^2*x2 ^3*x3^16*3^(1/2)*2^(1/2)+24876*x1^2*x2^2*x3^17*3^(1/2)*2^(1/2)+691*x1^2*x2*x3^ 18*3^(1/2)*2^(1/2)+691*x1*x2^18*x3^2*3^(1/2)*2^(1/2)+12438*x1*x2^17*x3^3*3^(1/2 )*2^(1/2)+103448*x1*x2^16*x3^4*3^(1/2)*2^(1/2)+527456*x1*x2^15*x3^5*3^(1/2)*2^( 1/2)+1844463*x1*x2^14*x3^6*3^(1/2)*2^(1/2)+4688530*x1*x2^13*x3^7*3^(1/2)*2^(1/2 )+8958352*x1*x2^12*x3^8*3^(1/2)*2^(1/2)+13120536*x1*x2^11*x3^9*3^(1/2)*2^(1/2)+ 14884012*x1*x2^10*x3^10*3^(1/2)*2^(1/2)+13120536*x1*x2^9*x3^11*3^(1/2)*2^(1/2)+ 8958352*x1*x2^8*x3^12*3^(1/2)*2^(1/2)+4688530*x1*x2^7*x3^13*3^(1/2)*2^(1/2)+ 1844463*x1*x2^6*x3^14*3^(1/2)*2^(1/2)+527456*x1*x2^5*x3^15*3^(1/2)*2^(1/2)+ 103448*x1*x2^4*x3^16*3^(1/2)*2^(1/2)+12438*x1*x2^3*x3^17*3^(1/2)*2^(1/2)+691*x1 *x2^2*x3^18*3^(1/2)*2^(1/2)+300677832*x1^11*x2^5*x3^5*3^(1/2)*2^(1/2)+256108944 *x1^11*x2^4*x3^6*3^(1/2)*2^(1/2)+147904536*x1^11*x2^3*x3^7*3^(1/2)*2^(1/2)+ 55255044*x1^11*x2^2*x3^8*3^(1/2)*2^(1/2)+13120536*x1^11*x2*x3^9*3^(1/2)*2^(1/2) +14884012*x1^10*x2^10*x3*3^(1/2)*2^(1/2)+67844348*x1^10*x2^9*x3^2*3^(1/2)*2^(1/ 2)+213120622*x1^10*x2^8*x3^3*3^(1/2)*2^(1/2)+397934823*x1^10*x2^7*x3^4*3^(1/2)* 2^(1/2)+522039518*x1^10*x2^6*x3^5*3^(1/2)*2^(1/2)+522039518*x1^10*x2^5*x3^6*3^( 1/2)*2^(1/2)+397934823*x1^10*x2^4*x3^7*3^(1/2)*2^(1/2)+213120622*x1^10*x2^3*x3^ 8*3^(1/2)*2^(1/2)+67844348*x1^10*x2^2*x3^9*3^(1/2)*2^(1/2)+14884012*x1^10*x2*x3 ^10*3^(1/2)*2^(1/2)+13120536*x1^9*x2^11*x3*3^(1/2)*2^(1/2)+67844348*x1^9*x2^10* x3^2*3^(1/2)*2^(1/2)+241378280*x1^9*x2^9*x3^3*3^(1/2)*2^(1/2)+477741550*x1^9*x2 ^8*x3^4*3^(1/2)*2^(1/2)+792914408*x1^9*x2^7*x3^5*3^(1/2)*2^(1/2)+1003654652*x1^ 9*x2^6*x3^6*3^(1/2)*2^(1/2)+792914408*x1^9*x2^5*x3^7*3^(1/2)*2^(1/2)+477741550* 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x2^3*x3^13*3^(1/2)*2^(1/2)+6712899*x1^5*x2^2*x3^14*3^(1/2)*2^(1/2)+527456*x1^5* x2*x3^15*3^(1/2)*2^(1/2)+103448*x1^4*x2^16*x3*3^(1/2)*2^(1/2)+1736724*x1^4*x2^ 15*x3^2*3^(1/2)*2^(1/2)+11369835*x1^4*x2^14*x3^3*3^(1/2)*2^(1/2)+44122820*x1^4* x2^13*x3^4*3^(1/2)*2^(1/2)+122506384*x1^4*x2^12*x3^5*3^(1/2)*2^(1/2)+256108944* x1^4*x2^11*x3^6*3^(1/2)*2^(1/2)+397934823*x1^4*x2^10*x3^7*3^(1/2)*2^(1/2)+ 477741550*x1^4*x2^9*x3^8*3^(1/2)*2^(1/2)+477741550*x1^4*x2^8*x3^9*3^(1/2)*2^(1/ 2)+397934823*x1^4*x2^7*x3^10*3^(1/2)*2^(1/2)+256108944*x1^4*x2^6*x3^11*3^(1/2)* 2^(1/2)+122506384*x1^4*x2^5*x3^12*3^(1/2)*2^(1/2)+44122820*x1^4*x2^4*x3^13*3^(1 /2)*2^(1/2)+11369835*x1^4*x2^3*x3^14*3^(1/2)*2^(1/2)+1736724*x1^4*x2^2*x3^15*3^ (1/2)*2^(1/2)+103448*x1^4*x2*x3^16*3^(1/2)*2^(1/2)+12438*x1^3*x2^17*x3*3^(1/2)* 2^(1/2)+67844348*x1^9*x2^2*x3^10*3^(1/2)*2^(1/2)+13120536*x1^9*x2*x3^11*3^(1/2) *2^(1/2)+8958352*x1^8*x2^12*x3*3^(1/2)*2^(1/2)+55255044*x1^8*x2^11*x3^2*3^(1/2) 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*x3*(x1^2*x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2*x3+x2*x3^2)/(x1^2+2*x1*x2+ 2*x1*x3+x2^2+2*x2*x3+x3^2))^(7/2)*x1^3*x2^4+42042000*(x1*x2*x3*(x1^2*x2+x1^2*x3 +x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2*x3+x2*x3^2)/(x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2*x3 +x3^2))^(7/2)*x1^3*x3^4+25225200*(x1*x2*x3*(x1^2*x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+ x1*x3^2+x2^2*x3+x2*x3^2)/(x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2*x3+x3^2))^(7/2)*x1^2* x2^5+25225200*(x1*x2*x3*(x1^2*x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2*x3+x2* x3^2)/(x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2*x3+x3^2))^(7/2)*x1^2*x3^5+8408400*(x1*x2* x3*(x1^2*x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2*x3+x2*x3^2)/(x1^2+2*x1*x2+2 *x1*x3+x2^2+2*x2*x3+x3^2))^(7/2)*x1*x2^6+8408400*(x1*x2*x3*(x1^2*x2+x1^2*x3+x1* x2^2-3*x1*x2*x3+x1*x3^2+x2^2*x3+x2*x3^2)/(x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2*x3+x3^ 2))^(7/2)*x1*x3^6+8408400*(x1*x2*x3*(x1^2*x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+x1*x3^2 +x2^2*x3+x2*x3^2)/(x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2*x3+x3^2))^(7/2)*x2^6*x3+ 25225200*(x1*x2*x3*(x1^2*x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2*x3+x2*x3^2) /(x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2*x3+x3^2))^(7/2)*x2^5*x3^2+42042000*(x1*x2*x3*( x1^2*x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2*x3+x2*x3^2)/(x1^2+2*x1*x2+2*x1* x3+x2^2+2*x2*x3+x3^2))^(7/2)*x2^4*x3^3+42042000*(x1*x2*x3*(x1^2*x2+x1^2*x3+x1* x2^2-3*x1*x2*x3+x1*x3^2+x2^2*x3+x2*x3^2)/(x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2*x3+x3^ 2))^(7/2)*x2^3*x3^4+25225200*(x1*x2*x3*(x1^2*x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+x1* x3^2+x2^2*x3+x2*x3^2)/(x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2*x3+x3^2))^(7/2)*x2^2*x3^5 +8408400*(x1*x2*x3*(x1^2*x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2*x3+x2*x3^2) /(x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2*x3+x3^2))^(7/2)*x2*x3^6+1201200*(x1*x2*x3*(x1^ 2*x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2*x3+x2*x3^2)/(x1^2+2*x1*x2+2*x1*x3+ x2^2+2*x2*x3+x3^2))^(7/2)*x1^7+1201200*(x1*x2*x3*(x1^2*x2+x1^2*x3+x1*x2^2-3*x1* x2*x3+x1*x3^2+x2^2*x3+x2*x3^2)/(x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2*x3+x3^2))^(7/2)* x2^7+1201200*(x1*x2*x3*(x1^2*x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2*x3+x2* x3^2)/(x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2*x3+x3^2))^(7/2)*x3^7) The , 8, -th (straight) moment of the Duration is: 13 13 12 2 12 12 2 3 A B C (3780 A B + 3780 A C + 52920 A B + 105840 A B C + 52920 A C 11 3 11 2 11 2 11 3 + 331542 A B + 1007064 A B C + 1007064 A B C + 331542 A C 10 4 10 3 10 2 2 10 3 + 1226664 A B + 5055912 A B C + 7658496 A B C + 5055912 A B C 10 4 9 5 9 4 9 3 2 + 1226664 A C + 2979252 A B + 15684408 A B C + 32173344 A B C 9 2 3 9 4 9 5 8 6 + 32173344 A B C + 15684408 A B C + 2979252 A C + 4995000 A B 8 5 8 4 2 8 3 3 + 32378760 A B C + 84723840 A B C + 114680160 A B C 8 2 4 8 5 8 6 7 7 + 84723840 A B C + 32378760 A B C + 4995000 A C + 5920047 A B 7 6 7 5 2 7 4 3 + 46157688 A B C + 148606272 A B C + 256482288 A B C 7 3 4 7 2 5 7 6 + 256482288 A B C + 148606272 A B C + 46157688 A B C 7 7 6 8 6 7 6 6 2 + 5920047 A C + 4995000 A B + 46157688 A B C + 178702848 A B C 6 5 3 6 4 4 6 3 5 + 379404864 A B C + 483729408 A B C + 379404864 A B C 6 2 6 6 7 6 8 5 9 + 178702848 A B C + 46157688 A B C + 4995000 A C + 2979252 A B 5 8 5 7 2 5 6 3 + 32378760 A B C + 148606272 A B C + 379404864 A B C 5 5 4 5 4 5 5 3 6 + 595026432 A B C + 595026432 A B C + 379404864 A B C 5 2 7 5 8 5 9 4 10 + 148606272 A B C + 32378760 A B C + 2979252 A C + 1226664 A B 4 9 4 8 2 4 7 3 + 15684408 A B C + 84723840 A B C + 256482288 A B C 4 6 4 4 5 5 4 4 6 + 483729408 A B C + 595026432 A B C + 483729408 A B C 4 3 7 4 2 8 4 9 + 256482288 A B C + 84723840 A B C + 15684408 A B C 4 10 3 11 3 10 3 9 2 + 1226664 A C + 331542 A B + 5055912 A B C + 32173344 A B C 3 8 3 3 7 4 3 6 5 + 114680160 A B C + 256482288 A B C + 379404864 A B C 3 5 6 3 4 7 3 3 8 + 379404864 A B C + 256482288 A B C + 114680160 A B C 3 2 9 3 10 3 11 2 12 + 32173344 A B C + 5055912 A B C + 331542 A C + 52920 A B 2 11 2 10 2 2 9 3 + 1007064 A B C + 7658496 A B C + 32173344 A B C 2 8 4 2 7 5 2 6 6 + 84723840 A B C + 148606272 A B C + 178702848 A B C 2 5 7 2 4 8 2 3 9 + 148606272 A B C + 84723840 A B C + 32173344 A B C 2 2 10 2 11 2 12 13 + 7658496 A B C + 1007064 A B C + 52920 A C + 3780 A B 12 11 2 10 3 9 4 + 105840 A B C + 1007064 A B C + 5055912 A B C + 15684408 A B C 8 5 7 6 6 7 + 32378760 A B C + 46157688 A B C + 46157688 A B C 5 8 4 9 3 10 + 32378760 A B C + 15684408 A B C + 5055912 A B C 2 11 12 13 13 + 1007064 A B C + 105840 A B C + 3780 A C + 3780 B C 12 2 11 3 10 4 9 5 + 52920 B C + 331542 B C + 1226664 B C + 2979252 B C 8 6 7 7 6 8 5 9 + 4995000 B C + 5920047 B C + 4995000 B C + 2979252 B C 4 10 3 11 2 12 13 11 + 1226664 B C + 331542 B C + 52920 B C + 3780 B C - 87066 A B 11 10 2 10 10 2 - 87066 A C - 1044792 A B - 2089584 A B C - 1044792 A C 9 3 9 2 9 2 9 3 - 5459706 A B - 16665768 A B C - 16665768 A B C - 5459706 A C 8 4 8 3 8 2 2 - 16288020 A B - 68018580 A B C - 103461120 A B C 8 3 8 4 7 5 7 4 - 68018580 A B C - 16288020 A C - 30576798 A B - 165026484 A B C 7 3 2 7 2 3 7 4 - 342573840 A B C - 342573840 A B C - 165026484 A B C 7 5 6 6 6 5 6 4 2 - 30576798 A C - 37585296 A B - 253855728 A B C - 680182272 A B C 6 3 3 6 2 4 6 5 - 927823680 A B C - 680182272 A B C - 253855728 A B C 6 6 5 7 5 6 5 5 2 - 37585296 A C - 30576798 A B - 253855728 A B C - 850897152 A B C 5 4 3 5 3 4 5 2 5 - 1497720672 A B C - 1497720672 A B C - 850897152 A B C 5 6 5 7 4 8 4 7 - 253855728 A B C - 30576798 A C - 16288020 A B - 165026484 A B C 4 6 2 4 5 3 4 4 4 - 680182272 A B C - 1497720672 A B C - 1932553728 A B C 4 3 5 4 2 6 4 7 - 1497720672 A B C - 680182272 A B C - 165026484 A B C 4 8 3 9 3 8 3 7 2 - 16288020 A C - 5459706 A B - 68018580 A B C - 342573840 A B C 3 6 3 3 5 4 3 4 5 - 927823680 A B C - 1497720672 A B C - 1497720672 A B C 3 3 6 3 2 7 3 8 - 927823680 A B C - 342573840 A B C - 68018580 A B C 3 9 2 10 2 9 2 8 2 - 5459706 A C - 1044792 A B - 16665768 A B C - 103461120 A B C 2 7 3 2 6 4 2 5 5 - 342573840 A B C - 680182272 A B C - 850897152 A B C 2 4 6 2 3 7 2 2 8 - 680182272 A B C - 342573840 A B C - 103461120 A B C 2 9 2 10 11 10 - 16665768 A B C - 1044792 A C - 87066 A B - 2089584 A B C 9 2 8 3 7 4 - 16665768 A B C - 68018580 A B C - 165026484 A B C 6 5 5 6 4 7 - 253855728 A B C - 253855728 A B C - 165026484 A B C 3 8 2 9 10 11 - 68018580 A B C - 16665768 A B C - 2089584 A B C - 87066 A C 11 10 2 9 3 8 4 - 87066 B C - 1044792 B C - 5459706 B C - 16288020 B C 7 5 6 6 5 7 4 8 - 30576798 B C - 37585296 B C - 30576798 B C - 16288020 B C 3 9 2 10 11 9 9 - 5459706 B C - 1044792 B C - 87066 B C + 679770 A B + 679770 A C 8 2 8 8 2 7 3 + 6797700 A B + 13595400 A B C + 6797700 A C + 28346409 A B 7 2 7 2 7 3 6 4 + 87282468 A B C + 87282468 A B C + 28346409 A C + 63626472 A B 6 3 6 2 2 6 3 + 272451816 A B C + 417650688 A B C + 272451816 A B C 6 4 5 5 5 4 + 63626472 A C + 82931940 A B + 471488472 A B C 5 3 2 5 2 3 5 4 + 1002796704 A B C + 1002796704 A B C + 471488472 A B C 5 5 4 6 4 5 + 82931940 A C + 63626472 A B + 471488472 A B C 4 4 2 4 3 3 4 2 4 + 1331261568 A B C + 1846799136 A B C + 1331261568 A B C 4 5 4 6 3 7 3 6 + 471488472 A B C + 63626472 A C + 28346409 A B + 272451816 A B C 3 5 2 3 4 3 3 3 4 + 1002796704 A B C + 1846799136 A B C + 1846799136 A B C 3 2 5 3 6 3 7 2 8 + 1002796704 A B C + 272451816 A B C + 28346409 A C + 6797700 A B 2 7 2 6 2 2 5 3 + 87282468 A B C + 417650688 A B C + 1002796704 A B C 2 4 4 2 3 5 2 2 6 + 1331261568 A B C + 1002796704 A B C + 417650688 A B C 2 7 2 8 9 8 + 87282468 A B C + 6797700 A C + 679770 A B + 13595400 A B C 7 2 6 3 5 4 + 87282468 A B C + 272451816 A B C + 471488472 A B C 4 5 3 6 2 7 + 471488472 A B C + 272451816 A B C + 87282468 A B C 8 9 9 8 2 + 13595400 A B C + 679770 A C + 679770 B C + 6797700 B C 7 3 6 4 5 5 4 6 + 28346409 B C + 63626472 B C + 82931940 B C + 63626472 B C 3 7 2 8 9 7 + 28346409 B C + 6797700 B C + 679770 B C - 1969110 A B 7 6 2 6 6 2 - 1969110 A C - 15752880 A B - 31505760 A B C - 15752880 A C 5 3 5 2 5 2 5 3 - 48571380 A B - 152277840 A B C - 152277840 A B C - 48571380 A C 4 4 4 3 4 2 2 - 70887960 A B - 322934040 A B C - 504092160 A B C 4 3 4 4 3 5 3 4 - 322934040 A B C - 70887960 A C - 48571380 A B - 322934040 A B C 3 3 2 3 2 3 3 4 - 735134400 A B C - 735134400 A B C - 322934040 A B C 3 5 2 6 2 5 2 4 2 - 48571380 A C - 15752880 A B - 152277840 A B C - 504092160 A B C 2 3 3 2 2 4 2 5 - 735134400 A B C - 504092160 A B C - 152277840 A B C 2 6 7 6 5 2 - 15752880 A C - 1969110 A B - 31505760 A B C - 152277840 A B C 4 3 3 4 2 5 - 322934040 A B C - 322934040 A B C - 152277840 A B C 6 7 7 6 2 - 31505760 A B C - 1969110 A C - 1969110 B C - 15752880 B C 5 3 4 4 3 5 2 6 - 48571380 B C - 70887960 B C - 48571380 B C - 15752880 B C 7 5 5 4 2 - 1969110 B C + 1315314 A B + 1315314 A C + 7891884 A B 4 4 2 3 3 3 2 + 15783768 A B C + 7891884 A C + 15126111 A B + 49981932 A B C 3 2 3 3 2 4 2 3 + 49981932 A B C + 15126111 A C + 7891884 A B + 49981932 A B C 2 2 2 2 3 2 4 5 + 84180096 A B C + 49981932 A B C + 7891884 A C + 1315314 A B 4 3 2 2 3 4 + 15783768 A B C + 49981932 A B C + 49981932 A B C + 15783768 A B C 5 5 4 2 3 3 + 1315314 A C + 1315314 B C + 7891884 B C + 15126111 B C 2 4 5 3 3 2 2 + 7891884 B C + 1315314 B C + 131040 A B + 131040 A C + 524160 A B 2 2 2 3 2 + 1048320 A B C + 524160 A C + 131040 A B + 1048320 A B C 2 3 3 2 2 3 + 1048320 A B C + 131040 A C + 131040 B C + 524160 B C + 131040 B C + 30848 A B + 30848 A C + 30848 B C - 8960)/(3203200 (A + B + C)) and in Maple notation 3/3203200*A*B*C*(3780*A^13*B+3780*A^13*C+52920*A^12*B^2+105840*A^12*B*C+52920*A ^12*C^2+331542*A^11*B^3+1007064*A^11*B^2*C+1007064*A^11*B*C^2+331542*A^11*C^3+ 1226664*A^10*B^4+5055912*A^10*B^3*C+7658496*A^10*B^2*C^2+5055912*A^10*B*C^3+ 1226664*A^10*C^4+2979252*A^9*B^5+15684408*A^9*B^4*C+32173344*A^9*B^3*C^2+ 32173344*A^9*B^2*C^3+15684408*A^9*B*C^4+2979252*A^9*C^5+4995000*A^8*B^6+ 32378760*A^8*B^5*C+84723840*A^8*B^4*C^2+114680160*A^8*B^3*C^3+84723840*A^8*B^2* C^4+32378760*A^8*B*C^5+4995000*A^8*C^6+5920047*A^7*B^7+46157688*A^7*B^6*C+ 148606272*A^7*B^5*C^2+256482288*A^7*B^4*C^3+256482288*A^7*B^3*C^4+148606272*A^7 *B^2*C^5+46157688*A^7*B*C^6+5920047*A^7*C^7+4995000*A^6*B^8+46157688*A^6*B^7*C+ 178702848*A^6*B^6*C^2+379404864*A^6*B^5*C^3+483729408*A^6*B^4*C^4+379404864*A^6 *B^3*C^5+178702848*A^6*B^2*C^6+46157688*A^6*B*C^7+4995000*A^6*C^8+2979252*A^5*B ^9+32378760*A^5*B^8*C+148606272*A^5*B^7*C^2+379404864*A^5*B^6*C^3+595026432*A^5 *B^5*C^4+595026432*A^5*B^4*C^5+379404864*A^5*B^3*C^6+148606272*A^5*B^2*C^7+ 32378760*A^5*B*C^8+2979252*A^5*C^9+1226664*A^4*B^10+15684408*A^4*B^9*C+84723840 *A^4*B^8*C^2+256482288*A^4*B^7*C^3+483729408*A^4*B^6*C^4+595026432*A^4*B^5*C^5+ 483729408*A^4*B^4*C^6+256482288*A^4*B^3*C^7+84723840*A^4*B^2*C^8+15684408*A^4*B *C^9+1226664*A^4*C^10+331542*A^3*B^11+5055912*A^3*B^10*C+32173344*A^3*B^9*C^2+ 114680160*A^3*B^8*C^3+256482288*A^3*B^7*C^4+379404864*A^3*B^6*C^5+379404864*A^3 *B^5*C^6+256482288*A^3*B^4*C^7+114680160*A^3*B^3*C^8+32173344*A^3*B^2*C^9+ 5055912*A^3*B*C^10+331542*A^3*C^11+52920*A^2*B^12+1007064*A^2*B^11*C+7658496*A^ 2*B^10*C^2+32173344*A^2*B^9*C^3+84723840*A^2*B^8*C^4+148606272*A^2*B^7*C^5+ 178702848*A^2*B^6*C^6+148606272*A^2*B^5*C^7+84723840*A^2*B^4*C^8+32173344*A^2*B ^3*C^9+7658496*A^2*B^2*C^10+1007064*A^2*B*C^11+52920*A^2*C^12+3780*A*B^13+ 105840*A*B^12*C+1007064*A*B^11*C^2+5055912*A*B^10*C^3+15684408*A*B^9*C^4+ 32378760*A*B^8*C^5+46157688*A*B^7*C^6+46157688*A*B^6*C^7+32378760*A*B^5*C^8+ 15684408*A*B^4*C^9+5055912*A*B^3*C^10+1007064*A*B^2*C^11+105840*A*B*C^12+3780*A *C^13+3780*B^13*C+52920*B^12*C^2+331542*B^11*C^3+1226664*B^10*C^4+2979252*B^9*C ^5+4995000*B^8*C^6+5920047*B^7*C^7+4995000*B^6*C^8+2979252*B^5*C^9+1226664*B^4* C^10+331542*B^3*C^11+52920*B^2*C^12+3780*B*C^13-87066*A^11*B-87066*A^11*C-\ 1044792*A^10*B^2-2089584*A^10*B*C-1044792*A^10*C^2-5459706*A^9*B^3-16665768*A^9 *B^2*C-16665768*A^9*B*C^2-5459706*A^9*C^3-16288020*A^8*B^4-68018580*A^8*B^3*C-\ 103461120*A^8*B^2*C^2-68018580*A^8*B*C^3-16288020*A^8*C^4-30576798*A^7*B^5-\ 165026484*A^7*B^4*C-342573840*A^7*B^3*C^2-342573840*A^7*B^2*C^3-165026484*A^7*B *C^4-30576798*A^7*C^5-37585296*A^6*B^6-253855728*A^6*B^5*C-680182272*A^6*B^4*C^ 2-927823680*A^6*B^3*C^3-680182272*A^6*B^2*C^4-253855728*A^6*B*C^5-37585296*A^6* C^6-30576798*A^5*B^7-253855728*A^5*B^6*C-850897152*A^5*B^5*C^2-1497720672*A^5*B ^4*C^3-1497720672*A^5*B^3*C^4-850897152*A^5*B^2*C^5-253855728*A^5*B*C^6-\ 30576798*A^5*C^7-16288020*A^4*B^8-165026484*A^4*B^7*C-680182272*A^4*B^6*C^2-\ 1497720672*A^4*B^5*C^3-1932553728*A^4*B^4*C^4-1497720672*A^4*B^3*C^5-680182272* A^4*B^2*C^6-165026484*A^4*B*C^7-16288020*A^4*C^8-5459706*A^3*B^9-68018580*A^3*B ^8*C-342573840*A^3*B^7*C^2-927823680*A^3*B^6*C^3-1497720672*A^3*B^5*C^4-\ 1497720672*A^3*B^4*C^5-927823680*A^3*B^3*C^6-342573840*A^3*B^2*C^7-68018580*A^3 *B*C^8-5459706*A^3*C^9-1044792*A^2*B^10-16665768*A^2*B^9*C-103461120*A^2*B^8*C^ 2-342573840*A^2*B^7*C^3-680182272*A^2*B^6*C^4-850897152*A^2*B^5*C^5-680182272*A ^2*B^4*C^6-342573840*A^2*B^3*C^7-103461120*A^2*B^2*C^8-16665768*A^2*B*C^9-\ 1044792*A^2*C^10-87066*A*B^11-2089584*A*B^10*C-16665768*A*B^9*C^2-68018580*A*B^ 8*C^3-165026484*A*B^7*C^4-253855728*A*B^6*C^5-253855728*A*B^5*C^6-165026484*A*B ^4*C^7-68018580*A*B^3*C^8-16665768*A*B^2*C^9-2089584*A*B*C^10-87066*A*C^11-\ 87066*B^11*C-1044792*B^10*C^2-5459706*B^9*C^3-16288020*B^8*C^4-30576798*B^7*C^5 -37585296*B^6*C^6-30576798*B^5*C^7-16288020*B^4*C^8-5459706*B^3*C^9-1044792*B^2 *C^10-87066*B*C^11+679770*A^9*B+679770*A^9*C+6797700*A^8*B^2+13595400*A^8*B*C+ 6797700*A^8*C^2+28346409*A^7*B^3+87282468*A^7*B^2*C+87282468*A^7*B*C^2+28346409 *A^7*C^3+63626472*A^6*B^4+272451816*A^6*B^3*C+417650688*A^6*B^2*C^2+272451816*A ^6*B*C^3+63626472*A^6*C^4+82931940*A^5*B^5+471488472*A^5*B^4*C+1002796704*A^5*B ^3*C^2+1002796704*A^5*B^2*C^3+471488472*A^5*B*C^4+82931940*A^5*C^5+63626472*A^4 *B^6+471488472*A^4*B^5*C+1331261568*A^4*B^4*C^2+1846799136*A^4*B^3*C^3+ 1331261568*A^4*B^2*C^4+471488472*A^4*B*C^5+63626472*A^4*C^6+28346409*A^3*B^7+ 272451816*A^3*B^6*C+1002796704*A^3*B^5*C^2+1846799136*A^3*B^4*C^3+1846799136*A^ 3*B^3*C^4+1002796704*A^3*B^2*C^5+272451816*A^3*B*C^6+28346409*A^3*C^7+6797700*A ^2*B^8+87282468*A^2*B^7*C+417650688*A^2*B^6*C^2+1002796704*A^2*B^5*C^3+ 1331261568*A^2*B^4*C^4+1002796704*A^2*B^3*C^5+417650688*A^2*B^2*C^6+87282468*A^ 2*B*C^7+6797700*A^2*C^8+679770*A*B^9+13595400*A*B^8*C+87282468*A*B^7*C^2+ 272451816*A*B^6*C^3+471488472*A*B^5*C^4+471488472*A*B^4*C^5+272451816*A*B^3*C^6 +87282468*A*B^2*C^7+13595400*A*B*C^8+679770*A*C^9+679770*B^9*C+6797700*B^8*C^2+ 28346409*B^7*C^3+63626472*B^6*C^4+82931940*B^5*C^5+63626472*B^4*C^6+28346409*B^ 3*C^7+6797700*B^2*C^8+679770*B*C^9-1969110*A^7*B-1969110*A^7*C-15752880*A^6*B^2 -31505760*A^6*B*C-15752880*A^6*C^2-48571380*A^5*B^3-152277840*A^5*B^2*C-\ 152277840*A^5*B*C^2-48571380*A^5*C^3-70887960*A^4*B^4-322934040*A^4*B^3*C-\ 504092160*A^4*B^2*C^2-322934040*A^4*B*C^3-70887960*A^4*C^4-48571380*A^3*B^5-\ 322934040*A^3*B^4*C-735134400*A^3*B^3*C^2-735134400*A^3*B^2*C^3-322934040*A^3*B *C^4-48571380*A^3*C^5-15752880*A^2*B^6-152277840*A^2*B^5*C-504092160*A^2*B^4*C^ 2-735134400*A^2*B^3*C^3-504092160*A^2*B^2*C^4-152277840*A^2*B*C^5-15752880*A^2* C^6-1969110*A*B^7-31505760*A*B^6*C-152277840*A*B^5*C^2-322934040*A*B^4*C^3-\ 322934040*A*B^3*C^4-152277840*A*B^2*C^5-31505760*A*B*C^6-1969110*A*C^7-1969110* B^7*C-15752880*B^6*C^2-48571380*B^5*C^3-70887960*B^4*C^4-48571380*B^3*C^5-\ 15752880*B^2*C^6-1969110*B*C^7+1315314*A^5*B+1315314*A^5*C+7891884*A^4*B^2+ 15783768*A^4*B*C+7891884*A^4*C^2+15126111*A^3*B^3+49981932*A^3*B^2*C+49981932*A ^3*B*C^2+15126111*A^3*C^3+7891884*A^2*B^4+49981932*A^2*B^3*C+84180096*A^2*B^2*C ^2+49981932*A^2*B*C^3+7891884*A^2*C^4+1315314*A*B^5+15783768*A*B^4*C+49981932*A *B^3*C^2+49981932*A*B^2*C^3+15783768*A*B*C^4+1315314*A*C^5+1315314*B^5*C+ 7891884*B^4*C^2+15126111*B^3*C^3+7891884*B^2*C^4+1315314*B*C^5+131040*A^3*B+ 131040*A^3*C+524160*A^2*B^2+1048320*A^2*B*C+524160*A^2*C^2+131040*A*B^3+1048320 *A*B^2*C+1048320*A*B*C^2+131040*A*C^3+131040*B^3*C+524160*B^2*C^2+131040*B*C^3+ 30848*A*B+30848*A*C+30848*B*C-8960)/(A+B+C) Hence the, 8, -th moment about the mean is 20 20 19 2 19 19 2 3 A B C (3780 A B + 3780 A C + 79380 A B + 158760 A B C + 79380 A C 18 3 18 2 18 2 18 3 + 781362 A B + 2207268 A B C + 2207268 A B C + 781362 A C 17 4 17 3 17 2 2 + 4791078 A B + 16714026 A B C + 23845896 A B C 17 3 17 4 16 5 16 4 + 16714026 A B C + 4791078 A C + 20512782 A B + 82313280 A B C 16 3 2 16 2 3 16 4 + 146849310 A B C + 146849310 A B C + 82313280 A B C 16 5 15 6 15 5 + 20512782 A C + 65145258 A B + 288436554 A B C 15 4 2 15 3 3 15 2 4 + 604080954 A B C + 761579316 A B C + 604080954 A B C 15 5 15 6 14 7 + 288436554 A B C + 65145258 A C + 159124329 A B 14 6 14 5 2 14 4 3 + 759255372 A B C + 1812906900 A B C + 2726487918 A B C 14 3 4 14 2 5 14 6 + 2726487918 A B C + 1812906900 A B C + 759255372 A B C 14 7 13 8 13 7 + 159124329 A C + 305873991 A B + 1557142947 A B C 13 6 2 13 5 3 13 4 4 + 4193950896 A B C + 7306053390 A B C + 8726742900 A B C 13 3 5 13 2 6 13 7 + 7306053390 A B C + 4193950896 A B C + 1557142947 A B C 13 8 12 9 12 8 + 305873991 A C + 469497717 A B + 2553602490 A B C 12 7 2 12 6 3 12 5 4 + 7734682233 A B C + 15245671482 A B C + 20985315246 A B C 12 4 5 12 3 6 12 2 7 + 20985315246 A B C + 15245671482 A B C + 7734682233 A B C 12 8 12 9 11 10 + 2553602490 A B C + 469497717 A C + 580485555 A B 11 9 11 8 2 11 7 3 + 3410369937 A B C + 11573374293 A B C + 25276922541 A B C 11 6 4 11 5 5 11 4 6 + 39343714488 A B C + 45620563716 A B C + 39343714488 A B C 11 3 7 11 2 8 11 9 + 25276922541 A B C + 11573374293 A B C + 3410369937 A B C 11 10 10 11 10 10 + 580485555 A C + 580485555 A B + 3750880068 A B C 10 9 2 10 8 3 10 7 4 + 14145728454 A B C + 33858949644 A B C + 59680265697 A B C 10 6 5 10 5 6 10 4 7 + 79314437124 A B C + 79314437124 A B C + 59680265697 A B C 10 3 8 10 2 9 10 10 + 33858949644 A B C + 14145728454 A B C + 3750880068 A B C 10 11 9 12 9 11 + 580485555 A C + 469497717 A B + 3410369937 A B C 9 10 2 9 9 3 9 8 4 + 14145728454 A B C + 37234940886 A B C + 73970310843 A B C 9 7 5 9 6 6 9 5 7 + 107293561947 A B C + 118706671512 A B C + 107293561947 A B C 9 4 8 9 3 9 9 2 10 + 73970310843 A B C + 37234940886 A B C + 14145728454 A B C 9 11 9 12 8 13 + 3410369937 A B C + 469497717 A C + 305873991 A B 8 12 8 11 2 8 10 3 + 2553602490 A B C + 11573374293 A B C + 33858949644 A B C 8 9 4 8 8 5 8 7 6 + 73970310843 A B C + 116750887110 A B C + 156248697033 A B C 8 6 7 8 5 8 8 4 9 + 156248697033 A B C + 116750887110 A B C + 73970310843 A B C 8 3 10 8 2 11 8 12 + 33858949644 A B C + 11573374293 A B C + 2553602490 A B C 8 13 7 14 7 13 + 305873991 A C + 159124329 A B + 1557142947 A B C 7 12 2 7 11 3 7 10 4 + 7734682233 A B C + 25276922541 A B C + 59680265697 A B C 7 9 5 7 8 6 7 7 7 + 107293561947 A B C + 156248697033 A B C + 157064402781 A B C 7 6 8 7 5 9 7 4 10 + 156248697033 A B C + 107293561947 A B C + 59680265697 A B C 7 3 11 7 2 12 7 13 + 25276922541 A B C + 7734682233 A B C + 1557142947 A B C 7 14 6 15 6 14 + 159124329 A C + 65145258 A B + 759255372 A B C 6 13 2 6 12 3 6 11 4 + 4193950896 A B C + 15245671482 A B C + 39343714488 A B C 6 10 5 6 9 6 6 8 7 + 79314437124 A B C + 118706671512 A B C + 156248697033 A B C 6 7 8 6 6 9 6 5 10 + 156248697033 A B C + 118706671512 A B C + 79314437124 A B C 6 4 11 6 3 12 6 2 13 + 39343714488 A B C + 15245671482 A B C + 4193950896 A B C 6 14 6 15 5 16 + 759255372 A B C + 65145258 A C + 20512782 A B 5 15 5 14 2 5 13 3 + 288436554 A B C + 1812906900 A B C + 7306053390 A B C 5 12 4 5 11 5 5 10 6 + 20985315246 A B C + 45620563716 A B C + 79314437124 A B C 5 9 7 5 8 8 5 7 9 + 107293561947 A B C + 116750887110 A B C + 107293561947 A B C 5 6 10 5 5 11 5 4 12 + 79314437124 A B C + 45620563716 A B C + 20985315246 A B C 5 3 13 5 2 14 5 15 + 7306053390 A B C + 1812906900 A B C + 288436554 A B C 5 16 4 17 4 16 + 20512782 A C + 4791078 A B + 82313280 A B C 4 15 2 4 14 3 4 13 4 + 604080954 A B C + 2726487918 A B C + 8726742900 A B C 4 12 5 4 11 6 4 10 7 + 20985315246 A B C + 39343714488 A B C + 59680265697 A B C 4 9 8 4 8 9 4 7 10 + 73970310843 A B C + 73970310843 A B C + 59680265697 A B C 4 6 11 4 5 12 4 4 13 + 39343714488 A B C + 20985315246 A B C + 8726742900 A B C 4 3 14 4 2 15 4 16 + 2726487918 A B C + 604080954 A B C + 82313280 A B C 4 17 3 18 3 17 3 16 2 + 4791078 A C + 781362 A B + 16714026 A B C + 146849310 A B C 3 15 3 3 14 4 3 13 5 + 761579316 A B C + 2726487918 A B C + 7306053390 A B C 3 12 6 3 11 7 3 10 8 + 15245671482 A B C + 25276922541 A B C + 33858949644 A B C 3 9 9 3 8 10 3 7 11 + 37234940886 A B C + 33858949644 A B C + 25276922541 A B C 3 6 12 3 5 13 3 4 14 + 15245671482 A B C + 7306053390 A B C + 2726487918 A B C 3 3 15 3 2 16 3 17 + 761579316 A B C + 146849310 A B C + 16714026 A B C 3 18 2 19 2 18 2 17 2 + 781362 A C + 79380 A B + 2207268 A B C + 23845896 A B C 2 16 3 2 15 4 2 14 5 + 146849310 A B C + 604080954 A B C + 1812906900 A B C 2 13 6 2 12 7 2 11 8 + 4193950896 A B C + 7734682233 A B C + 11573374293 A B C 2 10 9 2 9 10 2 8 11 + 14145728454 A B C + 14145728454 A B C + 11573374293 A B C 2 7 12 2 6 13 2 5 14 + 7734682233 A B C + 4193950896 A B C + 1812906900 A B C 2 4 15 2 3 16 2 2 17 + 604080954 A B C + 146849310 A B C + 23845896 A B C 2 18 2 19 20 19 + 2207268 A B C + 79380 A C + 3780 A B + 158760 A B C 18 2 17 3 16 4 + 2207268 A B C + 16714026 A B C + 82313280 A B C 15 5 14 6 13 7 + 288436554 A B C + 759255372 A B C + 1557142947 A B C 12 8 11 9 10 10 + 2553602490 A B C + 3410369937 A B C + 3750880068 A B C 9 11 8 12 7 13 + 3410369937 A B C + 2553602490 A B C + 1557142947 A B C 6 14 5 15 4 16 + 759255372 A B C + 288436554 A B C + 82313280 A B C 3 17 2 18 19 20 + 16714026 A B C + 2207268 A B C + 158760 A B C + 3780 A C 20 19 2 18 3 17 4 + 3780 B C + 79380 B C + 781362 B C + 4791078 B C 16 5 15 6 14 7 13 8 + 20512782 B C + 65145258 B C + 159124329 B C + 305873991 B C 12 9 11 10 10 11 + 469497717 B C + 580485555 B C + 580485555 B C 9 12 8 13 7 14 6 15 + 469497717 B C + 305873991 B C + 159124329 B C + 65145258 B C 5 16 4 17 3 18 2 19 + 20512782 B C + 4791078 B C + 781362 B C + 79380 B C 20 18 18 17 2 + 3780 B C - 87066 A B - 87066 A C - 1654254 A B 17 17 2 16 3 16 2 - 3308508 A B C - 1654254 A C - 14601636 A B - 41143158 A B C 16 2 16 3 15 4 - 41143158 A B C - 14601636 A C - 79493904 A B 15 3 15 2 2 15 3 - 275674266 A B C - 392360724 A B C - 275674266 A B C 15 4 14 5 14 4 - 79493904 A C - 298861794 A B - 1188458334 A B C 14 3 2 14 2 3 14 4 - 2110363290 A B C - 2110363290 A B C - 1188458334 A B C 14 5 13 6 13 5 - 298861794 A C - 823157118 A B - 3607625448 A B C 13 4 2 13 3 3 13 2 4 - 7523979120 A B C - 9479021580 A B C - 7523979120 A B C 13 5 13 6 12 7 - 3607625448 A B C - 823157118 A C - 1719507132 A B 12 6 12 5 2 12 4 3 - 8146471914 A B C - 19471058712 A B C - 29347907400 A B C 12 3 4 12 2 5 12 6 - 29347907400 A B C - 19471058712 A B C - 8146471914 A B C 12 7 11 8 11 7 - 1719507132 A C - 2781939888 A B - 14202857214 A B C 11 6 2 11 5 3 11 4 4 - 38638246374 A B C - 67680744768 A B C - 80926831440 A B C 11 3 5 11 2 6 11 7 - 67680744768 A B C - 38638246374 A B C - 14202857214 A B C 11 8 10 9 10 8 - 2781939888 A C - 3528573048 A B - 19616681448 A B C 10 7 2 10 6 3 10 5 4 - 60528054132 A B C - 119990332098 A B C - 165378029544 A B C 10 4 5 10 3 6 10 2 7 - 165378029544 A B C - 119990332098 A B C - 60528054132 A B C 10 8 10 9 9 10 - 19616681448 A B C - 3528573048 A C - 3528573048 A B 9 9 9 8 2 9 7 3 - 21806752968 A B C - 75676861260 A B C - 166825148490 A B C 9 6 4 9 5 5 9 4 6 - 262925556894 A B C - 306998179488 A B C - 262925556894 A B C 9 3 7 9 2 8 9 9 - 166825148490 A B C - 75676861260 A B C - 21806752968 A B C 9 10 8 11 8 10 - 3528573048 A C - 2781939888 A B - 19616681448 A B C 8 9 2 8 8 3 8 7 4 - 75676861260 A B C - 185594769360 A B C - 332749396980 A B C 8 6 5 8 5 6 8 4 7 - 438408516546 A B C - 438408516546 A B C - 332749396980 A B C 8 3 8 8 2 9 8 10 - 185594769360 A B C - 75676861260 A B C - 19616681448 A B C 8 11 7 12 7 11 - 2781939888 A C - 1719507132 A B - 14202857214 A B C 7 10 2 7 9 3 7 8 4 - 60528054132 A B C - 166825148490 A B C - 332749396980 A B C 7 7 5 7 6 6 7 5 7 - 490539833784 A B C - 560568196188 A B C - 490539833784 A B C 7 4 8 7 3 9 7 2 10 - 332749396980 A B C - 166825148490 A B C - 60528054132 A B C 7 11 7 12 6 13 - 14202857214 A B C - 1719507132 A C - 823157118 A B 6 12 6 11 2 6 10 3 - 8146471914 A B C - 38638246374 A B C - 119990332098 A B C 6 9 4 6 8 5 6 7 6 - 262925556894 A B C - 438408516546 A B C - 560568196188 A B C 6 6 7 6 5 8 6 4 9 - 560568196188 A B C - 438408516546 A B C - 262925556894 A B C 6 3 10 6 2 11 6 12 - 119990332098 A B C - 38638246374 A B C - 8146471914 A B C 6 13 5 14 5 13 - 823157118 A C - 298861794 A B - 3607625448 A B C 5 12 2 5 11 3 5 10 4 - 19471058712 A B C - 67680744768 A B C - 165378029544 A B C 5 9 5 5 8 6 5 7 7 - 306998179488 A B C - 438408516546 A B C - 490539833784 A B C 5 6 8 5 5 9 5 4 10 - 438408516546 A B C - 306998179488 A B C - 165378029544 A B C 5 3 11 5 2 12 5 13 - 67680744768 A B C - 19471058712 A B C - 3607625448 A B C 5 14 4 15 4 14 - 298861794 A C - 79493904 A B - 1188458334 A B C 4 13 2 4 12 3 4 11 4 - 7523979120 A B C - 29347907400 A B C - 80926831440 A B C 4 10 5 4 9 6 4 8 7 - 165378029544 A B C - 262925556894 A B C - 332749396980 A B C 4 7 8 4 6 9 4 5 10 - 332749396980 A B C - 262925556894 A B C - 165378029544 A B C 4 4 11 4 3 12 4 2 13 - 80926831440 A B C - 29347907400 A B C - 7523979120 A B C 4 14 4 15 3 16 - 1188458334 A B C - 79493904 A C - 14601636 A B 3 15 3 14 2 3 13 3 - 275674266 A B C - 2110363290 A B C - 9479021580 A B C 3 12 4 3 11 5 3 10 6 - 29347907400 A B C - 67680744768 A B C - 119990332098 A B C 3 9 7 3 8 8 3 7 9 - 166825148490 A B C - 185594769360 A B C - 166825148490 A B C 3 6 10 3 5 11 3 4 12 - 119990332098 A B C - 67680744768 A B C - 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799799040 A B C + 534279040 A B C + 135027328 A B C 3 6 2 7 2 6 2 5 2 + 647808 A C + 215936 A B + 54183680 A B C + 267895296 A B C 2 4 3 2 3 4 2 2 5 + 534279040 A B C + 534279040 A B C + 267895296 A B C 2 6 2 7 8 7 + 54183680 A B C + 215936 A C + 30848 A B + 9133440 A B C 6 2 5 3 4 4 + 54183680 A B C + 135027328 A B C + 179892480 A B C 3 5 2 6 7 8 + 135027328 A B C + 54183680 A B C + 9133440 A B C + 30848 A C 8 7 2 6 3 5 4 4 5 + 30848 B C + 215936 B C + 647808 B C + 1079680 B C + 1079680 B C 3 6 2 7 8 7 6 + 647808 B C + 215936 B C + 30848 B C - 8960 A - 62720 A B 6 5 2 5 5 2 4 3 - 62720 A C - 188160 A B - 376320 A B C - 188160 A C - 313600 A B 4 2 4 2 4 3 3 4 - 940800 A B C - 940800 A B C - 313600 A C - 313600 A B 3 3 3 2 2 3 3 3 4 - 1254400 A B C - 1881600 A B C - 1254400 A B C - 313600 A C 2 5 2 4 2 3 2 2 2 3 - 188160 A B - 940800 A B C - 1881600 A B C - 1881600 A B C 2 4 2 5 6 5 - 940800 A B C - 188160 A C - 62720 A B - 376320 A B C 4 2 3 3 2 4 5 - 940800 A B C - 1254400 A B C - 940800 A B C - 376320 A B C 6 7 6 5 2 4 3 - 62720 A C - 8960 B - 62720 B C - 188160 B C - 313600 B C 3 4 2 5 6 7 / - 313600 B C - 188160 B C - 62720 B C - 8960 C ) / (3203200 / 8 (A + B + C) ) and in Maple notation 3/3203200*A*B*C*(3780*A^20*B+3780*A^20*C+79380*A^19*B^2+158760*A^19*B*C+79380*A ^19*C^2+781362*A^18*B^3+2207268*A^18*B^2*C+2207268*A^18*B*C^2+781362*A^18*C^3+ 4791078*A^17*B^4+16714026*A^17*B^3*C+23845896*A^17*B^2*C^2+16714026*A^17*B*C^3+ 4791078*A^17*C^4+20512782*A^16*B^5+82313280*A^16*B^4*C+146849310*A^16*B^3*C^2+ 146849310*A^16*B^2*C^3+82313280*A^16*B*C^4+20512782*A^16*C^5+65145258*A^15*B^6+ 288436554*A^15*B^5*C+604080954*A^15*B^4*C^2+761579316*A^15*B^3*C^3+604080954*A^ 15*B^2*C^4+288436554*A^15*B*C^5+65145258*A^15*C^6+159124329*A^14*B^7+759255372* A^14*B^6*C+1812906900*A^14*B^5*C^2+2726487918*A^14*B^4*C^3+2726487918*A^14*B^3* C^4+1812906900*A^14*B^2*C^5+759255372*A^14*B*C^6+159124329*A^14*C^7+305873991*A 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*C^4-67680744768*A^11*B^3*C^5-38638246374*A^11*B^2*C^6-14202857214*A^11*B*C^7-\ 2781939888*A^11*C^8-3528573048*A^10*B^9-19616681448*A^10*B^8*C-60528054132*A^10 *B^7*C^2-119990332098*A^10*B^6*C^3-165378029544*A^10*B^5*C^4-165378029544*A^10* B^4*C^5-119990332098*A^10*B^3*C^6-60528054132*A^10*B^2*C^7-19616681448*A^10*B*C ^8-3528573048*A^10*C^9-3528573048*A^9*B^10-21806752968*A^9*B^9*C-75676861260*A^ 9*B^8*C^2-166825148490*A^9*B^7*C^3-262925556894*A^9*B^6*C^4-306998179488*A^9*B^ 5*C^5-262925556894*A^9*B^4*C^6-166825148490*A^9*B^3*C^7-75676861260*A^9*B^2*C^8 -21806752968*A^9*B*C^9-3528573048*A^9*C^10-2781939888*A^8*B^11-19616681448*A^8* B^10*C-75676861260*A^8*B^9*C^2-185594769360*A^8*B^8*C^3-332749396980*A^8*B^7*C^ 4-438408516546*A^8*B^6*C^5-438408516546*A^8*B^5*C^6-332749396980*A^8*B^4*C^7-\ 185594769360*A^8*B^3*C^8-75676861260*A^8*B^2*C^9-19616681448*A^8*B*C^10-\ 2781939888*A^8*C^11-1719507132*A^7*B^12-14202857214*A^7*B^11*C-60528054132*A^7* B^10*C^2-166825148490*A^7*B^9*C^3-332749396980*A^7*B^8*C^4-490539833784*A^7*B^7 *C^5-560568196188*A^7*B^6*C^6-490539833784*A^7*B^5*C^7-332749396980*A^7*B^4*C^8 -166825148490*A^7*B^3*C^9-60528054132*A^7*B^2*C^10-14202857214*A^7*B*C^11-\ 1719507132*A^7*C^12-823157118*A^6*B^13-8146471914*A^6*B^12*C-38638246374*A^6*B^ 11*C^2-119990332098*A^6*B^10*C^3-262925556894*A^6*B^9*C^4-438408516546*A^6*B^8* C^5-560568196188*A^6*B^7*C^6-560568196188*A^6*B^6*C^7-438408516546*A^6*B^5*C^8-\ 262925556894*A^6*B^4*C^9-119990332098*A^6*B^3*C^10-38638246374*A^6*B^2*C^11-\ 8146471914*A^6*B*C^12-823157118*A^6*C^13-298861794*A^5*B^14-3607625448*A^5*B^13 *C-19471058712*A^5*B^12*C^2-67680744768*A^5*B^11*C^3-165378029544*A^5*B^10*C^4-\ 306998179488*A^5*B^9*C^5-438408516546*A^5*B^8*C^6-490539833784*A^5*B^7*C^7-\ 438408516546*A^5*B^6*C^8-306998179488*A^5*B^5*C^9-165378029544*A^5*B^4*C^10-\ 67680744768*A^5*B^3*C^11-19471058712*A^5*B^2*C^12-3607625448*A^5*B*C^13-\ 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37427313924*A^6*B^3*C^4+23170607460*A^6*B^2*C^5+7669109448*A^6*B*C^6+1008188181 *A^6*C^7+696458763*A^5*B^8+6352813467*A^5*B^7*C+23170607460*A^5*B^6*C^2+ 45188459316*A^5*B^5*C^3+55348413120*A^5*B^4*C^4+45188459316*A^5*B^3*C^5+ 23170607460*A^5*B^2*C^6+6352813467*A^5*B*C^7+696458763*A^5*C^8+325540215*A^4*B^ 9+3572933364*A^4*B^8*C+15802840053*A^4*B^7*C^2+37427313924*A^4*B^6*C^3+ 55348413120*A^4*B^5*C^4+55348413120*A^4*B^4*C^5+37427313924*A^4*B^3*C^6+ 15802840053*A^4*B^2*C^7+3572933364*A^4*B*C^8+325540215*A^4*C^9+97990893*A^3*B^ 10+1316764449*A^3*B^9*C+7129677555*A^3*B^8*C^2+20743988265*A^3*B^7*C^3+ 37427313924*A^3*B^6*C^4+45188459316*A^3*B^5*C^5+37427313924*A^3*B^4*C^6+ 20743988265*A^3*B^3*C^7+7129677555*A^3*B^2*C^8+1316764449*A^3*B*C^9+97990893*A^ 3*C^10+17099082*A^2*B^11+294972678*A^2*B^10*C+1982448468*A^2*B^9*C^2+7129677555 *A^2*B^8*C^3+15802840053*A^2*B^7*C^4+23170607460*A^2*B^6*C^5+23170607460*A^2*B^ 5*C^6+15802840053*A^2*B^4*C^7+7129677555*A^2*B^3*C^8+1982448468*A^2*B^2*C^9+ 294972678*A^2*B*C^10+17099082*A^2*C^11+1315314*A*B^12+34198164*A*B^11*C+ 294972678*A*B^10*C^2+1316764449*A*B^9*C^3+3572933364*A*B^8*C^4+6352813467*A*B^7 *C^5+7669109448*A*B^6*C^6+6352813467*A*B^5*C^7+3572933364*A*B^4*C^8+1316764449* A*B^3*C^9+294972678*A*B^2*C^10+34198164*A*B*C^11+1315314*A*C^12+1315314*B^12*C+ 17099082*B^11*C^2+97990893*B^10*C^3+325540215*B^9*C^4+696458763*B^8*C^5+ 1008188181*B^7*C^6+1008188181*B^6*C^7+696458763*B^5*C^8+325540215*B^4*C^9+ 97990893*B^3*C^10+17099082*B^2*C^11+1315314*B*C^12+131040*A^10*B+131040*A^10*C+ 1441440*A^9*B^2+2882880*A^9*B*C+1441440*A^9*C^2+6552000*A^8*B^3+32366880*A^8*B^ 2*C+32366880*A^8*B*C^2+6552000*A^8*C^3+16511040*A^7*B^4+143095680*A^7*B^3*C+ 223554240*A^7*B^2*C^2+143095680*A^7*B*C^3+16511040*A^7*C^4+25683840*A^6*B^5+ 323930880*A^6*B^4*C+707878080*A^6*B^3*C^2+707878080*A^6*B^2*C^3+323930880*A^6*B *C^4+25683840*A^6*C^5+25683840*A^5*B^6+420900480*A^5*B^5*C+1219458240*A^5*B^4*C ^2+1648483200*A^5*B^3*C^3+1219458240*A^5*B^2*C^4+420900480*A^5*B*C^5+25683840*A ^5*C^6+16511040*A^4*B^7+323930880*A^4*B^6*C+1219458240*A^4*B^5*C^2+2154297600*A ^4*B^4*C^3+2154297600*A^4*B^3*C^4+1219458240*A^4*B^2*C^5+323930880*A^4*B*C^6+ 16511040*A^4*C^7+6552000*A^3*B^8+143095680*A^3*B^7*C+707878080*A^3*B^6*C^2+ 1648483200*A^3*B^5*C^3+2154297600*A^3*B^4*C^4+1648483200*A^3*B^3*C^5+707878080* A^3*B^2*C^6+143095680*A^3*B*C^7+6552000*A^3*C^8+1441440*A^2*B^9+32366880*A^2*B^ 8*C+223554240*A^2*B^7*C^2+707878080*A^2*B^6*C^3+1219458240*A^2*B^5*C^4+ 1219458240*A^2*B^4*C^5+707878080*A^2*B^3*C^6+223554240*A^2*B^2*C^7+32366880*A^2 *B*C^8+1441440*A^2*C^9+131040*A*B^10+2882880*A*B^9*C+32366880*A*B^8*C^2+ 143095680*A*B^7*C^3+323930880*A*B^6*C^4+420900480*A*B^5*C^5+323930880*A*B^4*C^6 +143095680*A*B^3*C^7+32366880*A*B^2*C^8+2882880*A*B*C^9+131040*A*C^10+131040*B^ 10*C+1441440*B^9*C^2+6552000*B^8*C^3+16511040*B^7*C^4+25683840*B^6*C^5+25683840 *B^5*C^6+16511040*B^4*C^7+6552000*B^3*C^8+1441440*B^2*C^9+131040*B*C^10+30848*A ^8*B+30848*A^8*C+215936*A^7*B^2+9133440*A^7*B*C+215936*A^7*C^2+647808*A^6*B^3+ 54183680*A^6*B^2*C+54183680*A^6*B*C^2+647808*A^6*C^3+1079680*A^5*B^4+135027328* A^5*B^3*C+267895296*A^5*B^2*C^2+135027328*A^5*B*C^3+1079680*A^5*C^4+1079680*A^4 *B^5+179892480*A^4*B^4*C+534279040*A^4*B^3*C^2+534279040*A^4*B^2*C^3+179892480* A^4*B*C^4+1079680*A^4*C^5+647808*A^3*B^6+135027328*A^3*B^5*C+534279040*A^3*B^4* C^2+799799040*A^3*B^3*C^3+534279040*A^3*B^2*C^4+135027328*A^3*B*C^5+647808*A^3* C^6+215936*A^2*B^7+54183680*A^2*B^6*C+267895296*A^2*B^5*C^2+534279040*A^2*B^4*C ^3+534279040*A^2*B^3*C^4+267895296*A^2*B^2*C^5+54183680*A^2*B*C^6+215936*A^2*C^ 7+30848*A*B^8+9133440*A*B^7*C+54183680*A*B^6*C^2+135027328*A*B^5*C^3+179892480* A*B^4*C^4+135027328*A*B^3*C^5+54183680*A*B^2*C^6+9133440*A*B*C^7+30848*A*C^8+ 30848*B^8*C+215936*B^7*C^2+647808*B^6*C^3+1079680*B^5*C^4+1079680*B^4*C^5+ 647808*B^3*C^6+215936*B^2*C^7+30848*B*C^8-8960*A^7-62720*A^6*B-62720*A^6*C-\ 188160*A^5*B^2-376320*A^5*B*C-188160*A^5*C^2-313600*A^4*B^3-940800*A^4*B^2*C-\ 940800*A^4*B*C^2-313600*A^4*C^3-313600*A^3*B^4-1254400*A^3*B^3*C-1881600*A^3*B^ 2*C^2-1254400*A^3*B*C^3-313600*A^3*C^4-188160*A^2*B^5-940800*A^2*B^4*C-1881600* A^2*B^3*C^2-1881600*A^2*B^2*C^3-940800*A^2*B*C^4-188160*A^2*C^5-62720*A*B^6-\ 376320*A*B^5*C-940800*A*B^4*C^2-1254400*A*B^3*C^3-940800*A*B^2*C^4-376320*A*B*C ^5-62720*A*C^6-8960*B^7-62720*B^6*C-188160*B^5*C^2-313600*B^4*C^3-313600*B^3*C^ 4-188160*B^2*C^5-62720*B*C^6-8960*C^7)/(A+B+C)^8 Hence the scaled, 8, -th moment about the mean is 20 20 19 2 19 19 2 (3780 A B + 3780 A C + 79380 A B + 158760 A B C + 79380 A C 18 3 18 2 18 2 18 3 + 781362 A B + 2207268 A B C + 2207268 A B C + 781362 A C 17 4 17 3 17 2 2 + 4791078 A B + 16714026 A B C + 23845896 A B C 17 3 17 4 16 5 16 4 + 16714026 A B C + 4791078 A C + 20512782 A B + 82313280 A B C 16 3 2 16 2 3 16 4 + 146849310 A B C + 146849310 A B C + 82313280 A B C 16 5 15 6 15 5 + 20512782 A C + 65145258 A B + 288436554 A B C 15 4 2 15 3 3 15 2 4 + 604080954 A B C + 761579316 A B C + 604080954 A B C 15 5 15 6 14 7 + 288436554 A B C + 65145258 A C + 159124329 A B 14 6 14 5 2 14 4 3 + 759255372 A B C + 1812906900 A B C + 2726487918 A B C 14 3 4 14 2 5 14 6 + 2726487918 A B C + 1812906900 A B C + 759255372 A B C 14 7 13 8 13 7 + 159124329 A C + 305873991 A B + 1557142947 A B C 13 6 2 13 5 3 13 4 4 + 4193950896 A B C + 7306053390 A B C + 8726742900 A B C 13 3 5 13 2 6 13 7 + 7306053390 A B C + 4193950896 A B C + 1557142947 A B C 13 8 12 9 12 8 + 305873991 A C + 469497717 A B + 2553602490 A B C 12 7 2 12 6 3 12 5 4 + 7734682233 A B C + 15245671482 A B C + 20985315246 A B C 12 4 5 12 3 6 12 2 7 + 20985315246 A B C + 15245671482 A B C + 7734682233 A B C 12 8 12 9 11 10 + 2553602490 A B C + 469497717 A C + 580485555 A B 11 9 11 8 2 11 7 3 + 3410369937 A B C + 11573374293 A B C + 25276922541 A B C 11 6 4 11 5 5 11 4 6 + 39343714488 A B C + 45620563716 A B C + 39343714488 A B C 11 3 7 11 2 8 11 9 + 25276922541 A B C + 11573374293 A B C + 3410369937 A B C 11 10 10 11 10 10 + 580485555 A C + 580485555 A B + 3750880068 A B C 10 9 2 10 8 3 10 7 4 + 14145728454 A B C + 33858949644 A B C + 59680265697 A B C 10 6 5 10 5 6 10 4 7 + 79314437124 A B C + 79314437124 A B C + 59680265697 A B C 10 3 8 10 2 9 10 10 + 33858949644 A B C + 14145728454 A B C + 3750880068 A B C 10 11 9 12 9 11 + 580485555 A C + 469497717 A B + 3410369937 A B C 9 10 2 9 9 3 9 8 4 + 14145728454 A B C + 37234940886 A B C + 73970310843 A B C 9 7 5 9 6 6 9 5 7 + 107293561947 A B C + 118706671512 A B C + 107293561947 A B C 9 4 8 9 3 9 9 2 10 + 73970310843 A B C + 37234940886 A B C + 14145728454 A B C 9 11 9 12 8 13 + 3410369937 A B C + 469497717 A C + 305873991 A B 8 12 8 11 2 8 10 3 + 2553602490 A B C + 11573374293 A B C + 33858949644 A B C 8 9 4 8 8 5 8 7 6 + 73970310843 A B C + 116750887110 A B C + 156248697033 A B C 8 6 7 8 5 8 8 4 9 + 156248697033 A B C + 116750887110 A B C + 73970310843 A B C 8 3 10 8 2 11 8 12 + 33858949644 A B C + 11573374293 A B C + 2553602490 A B C 8 13 7 14 7 13 + 305873991 A C + 159124329 A B + 1557142947 A B C 7 12 2 7 11 3 7 10 4 + 7734682233 A B C + 25276922541 A B C + 59680265697 A B C 7 9 5 7 8 6 7 7 7 + 107293561947 A B C + 156248697033 A B C + 157064402781 A B C 7 6 8 7 5 9 7 4 10 + 156248697033 A B C + 107293561947 A B C + 59680265697 A B C 7 3 11 7 2 12 7 13 + 25276922541 A B C + 7734682233 A B C + 1557142947 A B C 7 14 6 15 6 14 + 159124329 A C + 65145258 A B + 759255372 A B C 6 13 2 6 12 3 6 11 4 + 4193950896 A B C + 15245671482 A B C + 39343714488 A B C 6 10 5 6 9 6 6 8 7 + 79314437124 A B C + 118706671512 A B C + 156248697033 A B C 6 7 8 6 6 9 6 5 10 + 156248697033 A B C + 118706671512 A B C + 79314437124 A B C 6 4 11 6 3 12 6 2 13 + 39343714488 A B C + 15245671482 A B C + 4193950896 A B C 6 14 6 15 5 16 + 759255372 A B C + 65145258 A C + 20512782 A B 5 15 5 14 2 5 13 3 + 288436554 A B C + 1812906900 A B C + 7306053390 A B C 5 12 4 5 11 5 5 10 6 + 20985315246 A B C + 45620563716 A B C + 79314437124 A B C 5 9 7 5 8 8 5 7 9 + 107293561947 A B C + 116750887110 A B C + 107293561947 A B C 5 6 10 5 5 11 5 4 12 + 79314437124 A B C + 45620563716 A B C + 20985315246 A B C 5 3 13 5 2 14 5 15 + 7306053390 A B C + 1812906900 A B C + 288436554 A B C 5 16 4 17 4 16 + 20512782 A C + 4791078 A B + 82313280 A B C 4 15 2 4 14 3 4 13 4 + 604080954 A B C + 2726487918 A B C + 8726742900 A B C 4 12 5 4 11 6 4 10 7 + 20985315246 A B C + 39343714488 A B C + 59680265697 A B C 4 9 8 4 8 9 4 7 10 + 73970310843 A B C + 73970310843 A B C + 59680265697 A B C 4 6 11 4 5 12 4 4 13 + 39343714488 A B C + 20985315246 A B C + 8726742900 A B C 4 3 14 4 2 15 4 16 + 2726487918 A B C + 604080954 A B C + 82313280 A B C 4 17 3 18 3 17 3 16 2 + 4791078 A C + 781362 A B + 16714026 A B C + 146849310 A B C 3 15 3 3 14 4 3 13 5 + 761579316 A B C + 2726487918 A B C + 7306053390 A B C 3 12 6 3 11 7 3 10 8 + 15245671482 A B C + 25276922541 A B C + 33858949644 A B C 3 9 9 3 8 10 3 7 11 + 37234940886 A B C + 33858949644 A B C + 25276922541 A B C 3 6 12 3 5 13 3 4 14 + 15245671482 A B C + 7306053390 A B C + 2726487918 A B C 3 3 15 3 2 16 3 17 + 761579316 A B C + 146849310 A B C + 16714026 A B C 3 18 2 19 2 18 2 17 2 + 781362 A C + 79380 A B + 2207268 A B C + 23845896 A B C 2 16 3 2 15 4 2 14 5 + 146849310 A B C + 604080954 A B C + 1812906900 A B C 2 13 6 2 12 7 2 11 8 + 4193950896 A B C + 7734682233 A B C + 11573374293 A B C 2 10 9 2 9 10 2 8 11 + 14145728454 A B C + 14145728454 A B C + 11573374293 A B C 2 7 12 2 6 13 2 5 14 + 7734682233 A B C + 4193950896 A B C + 1812906900 A B C 2 4 15 2 3 16 2 2 17 + 604080954 A B C + 146849310 A B C + 23845896 A B C 2 18 2 19 20 19 + 2207268 A B C + 79380 A C + 3780 A B + 158760 A B C 18 2 17 3 16 4 + 2207268 A B C + 16714026 A B C + 82313280 A B C 15 5 14 6 13 7 + 288436554 A B C + 759255372 A B C + 1557142947 A B C 12 8 11 9 10 10 + 2553602490 A B C + 3410369937 A B C + 3750880068 A B C 9 11 8 12 7 13 + 3410369937 A B C + 2553602490 A B C + 1557142947 A B C 6 14 5 15 4 16 + 759255372 A B C + 288436554 A B C + 82313280 A B C 3 17 2 18 19 20 + 16714026 A B C + 2207268 A B C + 158760 A B C + 3780 A C 20 19 2 18 3 17 4 + 3780 B C + 79380 B C + 781362 B C + 4791078 B C 16 5 15 6 14 7 13 8 + 20512782 B C + 65145258 B C + 159124329 B C + 305873991 B C 12 9 11 10 10 11 + 469497717 B C + 580485555 B C + 580485555 B C 9 12 8 13 7 14 6 15 + 469497717 B C + 305873991 B C + 159124329 B C + 65145258 B C 5 16 4 17 3 18 2 19 + 20512782 B C + 4791078 B C + 781362 B C + 79380 B C 20 18 18 17 2 + 3780 B C - 87066 A B - 87066 A C - 1654254 A B 17 17 2 16 3 16 2 - 3308508 A B C - 1654254 A C - 14601636 A B - 41143158 A B C 16 2 16 3 15 4 - 41143158 A B C - 14601636 A C - 79493904 A B 15 3 15 2 2 15 3 - 275674266 A B C - 392360724 A B C - 275674266 A B C 15 4 14 5 14 4 - 79493904 A C - 298861794 A B - 1188458334 A B C 14 3 2 14 2 3 14 4 - 2110363290 A B C - 2110363290 A B C - 1188458334 A B C 14 5 13 6 13 5 - 298861794 A C - 823157118 A B - 3607625448 A B C 13 4 2 13 3 3 13 2 4 - 7523979120 A B C - 9479021580 A B C - 7523979120 A B C 13 5 13 6 12 7 - 3607625448 A B C - 823157118 A C - 1719507132 A B 12 6 12 5 2 12 4 3 - 8146471914 A B C - 19471058712 A B C - 29347907400 A B C 12 3 4 12 2 5 12 6 - 29347907400 A B C - 19471058712 A B C - 8146471914 A B C 12 7 11 8 11 7 - 1719507132 A C - 2781939888 A B - 14202857214 A B C 11 6 2 11 5 3 11 4 4 - 38638246374 A B C - 67680744768 A B C - 80926831440 A B C 11 3 5 11 2 6 11 7 - 67680744768 A B C - 38638246374 A B C - 14202857214 A B C 11 8 10 9 10 8 - 2781939888 A C - 3528573048 A B - 19616681448 A B C 10 7 2 10 6 3 10 5 4 - 60528054132 A B C - 119990332098 A B C - 165378029544 A B C 10 4 5 10 3 6 10 2 7 - 165378029544 A B C - 119990332098 A B C - 60528054132 A B C 10 8 10 9 9 10 - 19616681448 A B C - 3528573048 A C - 3528573048 A B 9 9 9 8 2 9 7 3 - 21806752968 A B C - 75676861260 A B C - 166825148490 A B C 9 6 4 9 5 5 9 4 6 - 262925556894 A B C - 306998179488 A B C - 262925556894 A B C 9 3 7 9 2 8 9 9 - 166825148490 A B C - 75676861260 A B C - 21806752968 A B C 9 10 8 11 8 10 - 3528573048 A C - 2781939888 A B - 19616681448 A B C 8 9 2 8 8 3 8 7 4 - 75676861260 A B C - 185594769360 A B C - 332749396980 A B C 8 6 5 8 5 6 8 4 7 - 438408516546 A B C - 438408516546 A B C - 332749396980 A B C 8 3 8 8 2 9 8 10 - 185594769360 A B C - 75676861260 A B C - 19616681448 A B C 8 11 7 12 7 11 - 2781939888 A C - 1719507132 A B - 14202857214 A B C 7 10 2 7 9 3 7 8 4 - 60528054132 A B C - 166825148490 A B C - 332749396980 A B C 7 7 5 7 6 6 7 5 7 - 490539833784 A B C - 560568196188 A B C - 490539833784 A B C 7 4 8 7 3 9 7 2 10 - 332749396980 A B C - 166825148490 A B C - 60528054132 A B C 7 11 7 12 6 13 - 14202857214 A B C - 1719507132 A C - 823157118 A B 6 12 6 11 2 6 10 3 - 8146471914 A B C - 38638246374 A B C - 119990332098 A B C 6 9 4 6 8 5 6 7 6 - 262925556894 A B C - 438408516546 A B C - 560568196188 A B C 6 6 7 6 5 8 6 4 9 - 560568196188 A B C - 438408516546 A B C - 262925556894 A B C 6 3 10 6 2 11 6 12 - 119990332098 A B C - 38638246374 A B C - 8146471914 A B C 6 13 5 14 5 13 - 823157118 A C - 298861794 A B - 3607625448 A B C 5 12 2 5 11 3 5 10 4 - 19471058712 A B C - 67680744768 A B C - 165378029544 A B C 5 9 5 5 8 6 5 7 7 - 306998179488 A B C - 438408516546 A B C - 490539833784 A B C 5 6 8 5 5 9 5 4 10 - 438408516546 A B C - 306998179488 A B C - 165378029544 A B C 5 3 11 5 2 12 5 13 - 67680744768 A B C - 19471058712 A B C - 3607625448 A B C 5 14 4 15 4 14 - 298861794 A C - 79493904 A B - 1188458334 A B C 4 13 2 4 12 3 4 11 4 - 7523979120 A B C - 29347907400 A B C - 80926831440 A B C 4 10 5 4 9 6 4 8 7 - 165378029544 A B C - 262925556894 A B C - 332749396980 A B C 4 7 8 4 6 9 4 5 10 - 332749396980 A B C - 262925556894 A B C - 165378029544 A B C 4 4 11 4 3 12 4 2 13 - 80926831440 A B C - 29347907400 A B C - 7523979120 A B C 4 14 4 15 3 16 - 1188458334 A B C - 79493904 A C - 14601636 A B 3 15 3 14 2 3 13 3 - 275674266 A B C - 2110363290 A B C - 9479021580 A B C 3 12 4 3 11 5 3 10 6 - 29347907400 A B C - 67680744768 A B C - 119990332098 A B C 3 9 7 3 8 8 3 7 9 - 166825148490 A B C - 185594769360 A B C - 166825148490 A B C 3 6 10 3 5 11 3 4 12 - 119990332098 A B C - 67680744768 A B C - 29347907400 A B C 3 3 13 3 2 14 3 15 - 9479021580 A B C - 2110363290 A B C - 275674266 A B C 3 16 2 17 2 16 - 14601636 A C - 1654254 A B - 41143158 A B C 2 15 2 2 14 3 2 13 4 - 392360724 A B C - 2110363290 A B C - 7523979120 A B C 2 12 5 2 11 6 2 10 7 - 19471058712 A B C - 38638246374 A B C - 60528054132 A B C 2 9 8 2 8 9 2 7 10 - 75676861260 A B C - 75676861260 A B C - 60528054132 A B C 2 6 11 2 5 12 2 4 13 - 38638246374 A B C - 19471058712 A B C - 7523979120 A B C 2 3 14 2 2 15 2 16 - 2110363290 A B C - 392360724 A B C - 41143158 A B C 2 17 18 17 16 2 - 1654254 A C - 87066 A B - 3308508 A B C - 41143158 A B C 15 3 14 4 13 5 - 275674266 A B C 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647808 A C 5 4 5 3 5 2 2 + 1079680 A B + 135027328 A B C + 267895296 A B C 5 3 5 4 4 5 4 4 + 135027328 A B C + 1079680 A C + 1079680 A B + 179892480 A B C 4 3 2 4 2 3 4 4 + 534279040 A B C + 534279040 A B C + 179892480 A B C 4 5 3 6 3 5 3 4 2 + 1079680 A C + 647808 A B + 135027328 A B C + 534279040 A B C 3 3 3 3 2 4 3 5 + 799799040 A B C + 534279040 A B C + 135027328 A B C 3 6 2 7 2 6 2 5 2 + 647808 A C + 215936 A B + 54183680 A B C + 267895296 A B C 2 4 3 2 3 4 2 2 5 + 534279040 A B C + 534279040 A B C + 267895296 A B C 2 6 2 7 8 7 + 54183680 A B C + 215936 A C + 30848 A B + 9133440 A B C 6 2 5 3 4 4 + 54183680 A B C + 135027328 A B C + 179892480 A B C 3 5 2 6 7 8 + 135027328 A B C + 54183680 A B C + 9133440 A B C + 30848 A C 8 7 2 6 3 5 4 4 5 + 30848 B C + 215936 B C + 647808 B C + 1079680 B C + 1079680 B C 3 6 2 7 8 7 6 + 647808 B C + 215936 B C + 30848 B C - 8960 A - 62720 A B 6 5 2 5 5 2 4 3 - 62720 A C - 188160 A B - 376320 A B C - 188160 A C - 313600 A B 4 2 4 2 4 3 3 4 - 940800 A B C - 940800 A B C - 313600 A C - 313600 A B 3 3 3 2 2 3 3 3 4 - 1254400 A B C - 1881600 A B C - 1254400 A B C - 313600 A C 2 5 2 4 2 3 2 2 2 3 - 188160 A B - 940800 A B C - 1881600 A B C - 1881600 A B C 2 4 2 5 6 5 - 940800 A B C - 188160 A C - 62720 A B - 376320 A B C 4 2 3 3 2 4 5 - 940800 A B C - 1254400 A B C - 940800 A B C - 376320 A B C 6 7 6 5 2 4 3 - 62720 A C - 8960 B - 62720 B C - 188160 B C - 313600 B C 3 4 2 5 6 7 / 3 3 - 313600 B C - 188160 B C - 62720 B C - 8960 C ) / (5405400 A B / 3 2 2 2 2 2 2 4 C (A B + A C + A B - 3 A B C + A C + B C + B C - A - B - C) ) and in Maple notation 1/5405400/A^3/B^3/C^3*(3780*A^20*B+3780*A^20*C+79380*A^19*B^2+158760*A^19*B*C+ 79380*A^19*C^2+781362*A^18*B^3+2207268*A^18*B^2*C+2207268*A^18*B*C^2+781362*A^ 18*C^3+4791078*A^17*B^4+16714026*A^17*B^3*C+23845896*A^17*B^2*C^2+16714026*A^17 *B*C^3+4791078*A^17*C^4+20512782*A^16*B^5+82313280*A^16*B^4*C+146849310*A^16*B^ 3*C^2+146849310*A^16*B^2*C^3+82313280*A^16*B*C^4+20512782*A^16*C^5+65145258*A^ 15*B^6+288436554*A^15*B^5*C+604080954*A^15*B^4*C^2+761579316*A^15*B^3*C^3+ 604080954*A^15*B^2*C^4+288436554*A^15*B*C^5+65145258*A^15*C^6+159124329*A^14*B^ 7+759255372*A^14*B^6*C+1812906900*A^14*B^5*C^2+2726487918*A^14*B^4*C^3+ 2726487918*A^14*B^3*C^4+1812906900*A^14*B^2*C^5+759255372*A^14*B*C^6+159124329* A^14*C^7+305873991*A^13*B^8+1557142947*A^13*B^7*C+4193950896*A^13*B^6*C^2+ 7306053390*A^13*B^5*C^3+8726742900*A^13*B^4*C^4+7306053390*A^13*B^3*C^5+ 4193950896*A^13*B^2*C^6+1557142947*A^13*B*C^7+305873991*A^13*C^8+469497717*A^12 *B^9+2553602490*A^12*B^8*C+7734682233*A^12*B^7*C^2+15245671482*A^12*B^6*C^3+ 20985315246*A^12*B^5*C^4+20985315246*A^12*B^4*C^5+15245671482*A^12*B^3*C^6+ 7734682233*A^12*B^2*C^7+2553602490*A^12*B*C^8+469497717*A^12*C^9+580485555*A^11 *B^10+3410369937*A^11*B^9*C+11573374293*A^11*B^8*C^2+25276922541*A^11*B^7*C^3+ 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37234940886*A^3*B^9*C^9+33858949644*A^3*B^8*C^10+25276922541*A^3*B^7*C^11+ 15245671482*A^3*B^6*C^12+7306053390*A^3*B^5*C^13+2726487918*A^3*B^4*C^14+ 761579316*A^3*B^3*C^15+146849310*A^3*B^2*C^16+16714026*A^3*B*C^17+781362*A^3*C^ 18+79380*A^2*B^19+2207268*A^2*B^18*C+23845896*A^2*B^17*C^2+146849310*A^2*B^16*C ^3+604080954*A^2*B^15*C^4+1812906900*A^2*B^14*C^5+4193950896*A^2*B^13*C^6+ 7734682233*A^2*B^12*C^7+11573374293*A^2*B^11*C^8+14145728454*A^2*B^10*C^9+ 14145728454*A^2*B^9*C^10+11573374293*A^2*B^8*C^11+7734682233*A^2*B^7*C^12+ 4193950896*A^2*B^6*C^13+1812906900*A^2*B^5*C^14+604080954*A^2*B^4*C^15+ 146849310*A^2*B^3*C^16+23845896*A^2*B^2*C^17+2207268*A^2*B*C^18+79380*A^2*C^19+ 3780*A*B^20+158760*A*B^19*C+2207268*A*B^18*C^2+16714026*A*B^17*C^3+82313280*A*B ^16*C^4+288436554*A*B^15*C^5+759255372*A*B^14*C^6+1557142947*A*B^13*C^7+ 2553602490*A*B^12*C^8+3410369937*A*B^11*C^9+3750880068*A*B^10*C^10+3410369937*A *B^9*C^11+2553602490*A*B^8*C^12+1557142947*A*B^7*C^13+759255372*A*B^6*C^14+ 288436554*A*B^5*C^15+82313280*A*B^4*C^16+16714026*A*B^3*C^17+2207268*A*B^2*C^18 +158760*A*B*C^19+3780*A*C^20+3780*B^20*C+79380*B^19*C^2+781362*B^18*C^3+4791078 *B^17*C^4+20512782*B^16*C^5+65145258*B^15*C^6+159124329*B^14*C^7+305873991*B^13 *C^8+469497717*B^12*C^9+580485555*B^11*C^10+580485555*B^10*C^11+469497717*B^9*C ^12+305873991*B^8*C^13+159124329*B^7*C^14+65145258*B^6*C^15+20512782*B^5*C^16+ 4791078*B^4*C^17+781362*B^3*C^18+79380*B^2*C^19+3780*B*C^20-87066*A^18*B-87066* A^18*C-1654254*A^17*B^2-3308508*A^17*B*C-1654254*A^17*C^2-14601636*A^16*B^3-\ 41143158*A^16*B^2*C-41143158*A^16*B*C^2-14601636*A^16*C^3-79493904*A^15*B^4-\ 275674266*A^15*B^3*C-392360724*A^15*B^2*C^2-275674266*A^15*B*C^3-79493904*A^15* C^4-298861794*A^14*B^5-1188458334*A^14*B^4*C-2110363290*A^14*B^3*C^2-2110363290 *A^14*B^2*C^3-1188458334*A^14*B*C^4-298861794*A^14*C^5-823157118*A^13*B^6-\ 3607625448*A^13*B^5*C-7523979120*A^13*B^4*C^2-9479021580*A^13*B^3*C^3-\ 7523979120*A^13*B^2*C^4-3607625448*A^13*B*C^5-823157118*A^13*C^6-1719507132*A^ 12*B^7-8146471914*A^12*B^6*C-19471058712*A^12*B^5*C^2-29347907400*A^12*B^4*C^3-\ 29347907400*A^12*B^3*C^4-19471058712*A^12*B^2*C^5-8146471914*A^12*B*C^6-\ 1719507132*A^12*C^7-2781939888*A^11*B^8-14202857214*A^11*B^7*C-38638246374*A^11 *B^6*C^2-67680744768*A^11*B^5*C^3-80926831440*A^11*B^4*C^4-67680744768*A^11*B^3 *C^5-38638246374*A^11*B^2*C^6-14202857214*A^11*B*C^7-2781939888*A^11*C^8-\ 3528573048*A^10*B^9-19616681448*A^10*B^8*C-60528054132*A^10*B^7*C^2-\ 119990332098*A^10*B^6*C^3-165378029544*A^10*B^5*C^4-165378029544*A^10*B^4*C^5-\ 119990332098*A^10*B^3*C^6-60528054132*A^10*B^2*C^7-19616681448*A^10*B*C^8-\ 3528573048*A^10*C^9-3528573048*A^9*B^10-21806752968*A^9*B^9*C-75676861260*A^9*B ^8*C^2-166825148490*A^9*B^7*C^3-262925556894*A^9*B^6*C^4-306998179488*A^9*B^5*C ^5-262925556894*A^9*B^4*C^6-166825148490*A^9*B^3*C^7-75676861260*A^9*B^2*C^8-\ 21806752968*A^9*B*C^9-3528573048*A^9*C^10-2781939888*A^8*B^11-19616681448*A^8*B ^10*C-75676861260*A^8*B^9*C^2-185594769360*A^8*B^8*C^3-332749396980*A^8*B^7*C^4 -438408516546*A^8*B^6*C^5-438408516546*A^8*B^5*C^6-332749396980*A^8*B^4*C^7-\ 185594769360*A^8*B^3*C^8-75676861260*A^8*B^2*C^9-19616681448*A^8*B*C^10-\ 2781939888*A^8*C^11-1719507132*A^7*B^12-14202857214*A^7*B^11*C-60528054132*A^7* B^10*C^2-166825148490*A^7*B^9*C^3-332749396980*A^7*B^8*C^4-490539833784*A^7*B^7 *C^5-560568196188*A^7*B^6*C^6-490539833784*A^7*B^5*C^7-332749396980*A^7*B^4*C^8 -166825148490*A^7*B^3*C^9-60528054132*A^7*B^2*C^10-14202857214*A^7*B*C^11-\ 1719507132*A^7*C^12-823157118*A^6*B^13-8146471914*A^6*B^12*C-38638246374*A^6*B^ 11*C^2-119990332098*A^6*B^10*C^3-262925556894*A^6*B^9*C^4-438408516546*A^6*B^8* C^5-560568196188*A^6*B^7*C^6-560568196188*A^6*B^6*C^7-438408516546*A^6*B^5*C^8-\ 262925556894*A^6*B^4*C^9-119990332098*A^6*B^3*C^10-38638246374*A^6*B^2*C^11-\ 8146471914*A^6*B*C^12-823157118*A^6*C^13-298861794*A^5*B^14-3607625448*A^5*B^13 *C-19471058712*A^5*B^12*C^2-67680744768*A^5*B^11*C^3-165378029544*A^5*B^10*C^4-\ 306998179488*A^5*B^9*C^5-438408516546*A^5*B^8*C^6-490539833784*A^5*B^7*C^7-\ 438408516546*A^5*B^6*C^8-306998179488*A^5*B^5*C^9-165378029544*A^5*B^4*C^10-\ 67680744768*A^5*B^3*C^11-19471058712*A^5*B^2*C^12-3607625448*A^5*B*C^13-\ 298861794*A^5*C^14-79493904*A^4*B^15-1188458334*A^4*B^14*C-7523979120*A^4*B^13* C^2-29347907400*A^4*B^12*C^3-80926831440*A^4*B^11*C^4-165378029544*A^4*B^10*C^5 -262925556894*A^4*B^9*C^6-332749396980*A^4*B^8*C^7-332749396980*A^4*B^7*C^8-\ 262925556894*A^4*B^6*C^9-165378029544*A^4*B^5*C^10-80926831440*A^4*B^4*C^11-\ 29347907400*A^4*B^3*C^12-7523979120*A^4*B^2*C^13-1188458334*A^4*B*C^14-79493904 *A^4*C^15-14601636*A^3*B^16-275674266*A^3*B^15*C-2110363290*A^3*B^14*C^2-\ 9479021580*A^3*B^13*C^3-29347907400*A^3*B^12*C^4-67680744768*A^3*B^11*C^5-\ 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20743988265*A^3*B^3*C^7+7129677555*A^3*B^2*C^8+1316764449*A^3*B*C^9+97990893*A^ 3*C^10+17099082*A^2*B^11+294972678*A^2*B^10*C+1982448468*A^2*B^9*C^2+7129677555 *A^2*B^8*C^3+15802840053*A^2*B^7*C^4+23170607460*A^2*B^6*C^5+23170607460*A^2*B^ 5*C^6+15802840053*A^2*B^4*C^7+7129677555*A^2*B^3*C^8+1982448468*A^2*B^2*C^9+ 294972678*A^2*B*C^10+17099082*A^2*C^11+1315314*A*B^12+34198164*A*B^11*C+ 294972678*A*B^10*C^2+1316764449*A*B^9*C^3+3572933364*A*B^8*C^4+6352813467*A*B^7 *C^5+7669109448*A*B^6*C^6+6352813467*A*B^5*C^7+3572933364*A*B^4*C^8+1316764449* A*B^3*C^9+294972678*A*B^2*C^10+34198164*A*B*C^11+1315314*A*C^12+1315314*B^12*C+ 17099082*B^11*C^2+97990893*B^10*C^3+325540215*B^9*C^4+696458763*B^8*C^5+ 1008188181*B^7*C^6+1008188181*B^6*C^7+696458763*B^5*C^8+325540215*B^4*C^9+ 97990893*B^3*C^10+17099082*B^2*C^11+1315314*B*C^12+131040*A^10*B+131040*A^10*C+ 1441440*A^9*B^2+2882880*A^9*B*C+1441440*A^9*C^2+6552000*A^8*B^3+32366880*A^8*B^ 2*C+32366880*A^8*B*C^2+6552000*A^8*C^3+16511040*A^7*B^4+143095680*A^7*B^3*C+ 223554240*A^7*B^2*C^2+143095680*A^7*B*C^3+16511040*A^7*C^4+25683840*A^6*B^5+ 323930880*A^6*B^4*C+707878080*A^6*B^3*C^2+707878080*A^6*B^2*C^3+323930880*A^6*B *C^4+25683840*A^6*C^5+25683840*A^5*B^6+420900480*A^5*B^5*C+1219458240*A^5*B^4*C ^2+1648483200*A^5*B^3*C^3+1219458240*A^5*B^2*C^4+420900480*A^5*B*C^5+25683840*A ^5*C^6+16511040*A^4*B^7+323930880*A^4*B^6*C+1219458240*A^4*B^5*C^2+2154297600*A ^4*B^4*C^3+2154297600*A^4*B^3*C^4+1219458240*A^4*B^2*C^5+323930880*A^4*B*C^6+ 16511040*A^4*C^7+6552000*A^3*B^8+143095680*A^3*B^7*C+707878080*A^3*B^6*C^2+ 1648483200*A^3*B^5*C^3+2154297600*A^3*B^4*C^4+1648483200*A^3*B^3*C^5+707878080* A^3*B^2*C^6+143095680*A^3*B*C^7+6552000*A^3*C^8+1441440*A^2*B^9+32366880*A^2*B^ 8*C+223554240*A^2*B^7*C^2+707878080*A^2*B^6*C^3+1219458240*A^2*B^5*C^4+ 1219458240*A^2*B^4*C^5+707878080*A^2*B^3*C^6+223554240*A^2*B^2*C^7+32366880*A^2 *B*C^8+1441440*A^2*C^9+131040*A*B^10+2882880*A*B^9*C+32366880*A*B^8*C^2+ 143095680*A*B^7*C^3+323930880*A*B^6*C^4+420900480*A*B^5*C^5+323930880*A*B^4*C^6 +143095680*A*B^3*C^7+32366880*A*B^2*C^8+2882880*A*B*C^9+131040*A*C^10+131040*B^ 10*C+1441440*B^9*C^2+6552000*B^8*C^3+16511040*B^7*C^4+25683840*B^6*C^5+25683840 *B^5*C^6+16511040*B^4*C^7+6552000*B^3*C^8+1441440*B^2*C^9+131040*B*C^10+30848*A ^8*B+30848*A^8*C+215936*A^7*B^2+9133440*A^7*B*C+215936*A^7*C^2+647808*A^6*B^3+ 54183680*A^6*B^2*C+54183680*A^6*B*C^2+647808*A^6*C^3+1079680*A^5*B^4+135027328* A^5*B^3*C+267895296*A^5*B^2*C^2+135027328*A^5*B*C^3+1079680*A^5*C^4+1079680*A^4 *B^5+179892480*A^4*B^4*C+534279040*A^4*B^3*C^2+534279040*A^4*B^2*C^3+179892480* A^4*B*C^4+1079680*A^4*C^5+647808*A^3*B^6+135027328*A^3*B^5*C+534279040*A^3*B^4* C^2+799799040*A^3*B^3*C^3+534279040*A^3*B^2*C^4+135027328*A^3*B*C^5+647808*A^3* C^6+215936*A^2*B^7+54183680*A^2*B^6*C+267895296*A^2*B^5*C^2+534279040*A^2*B^4*C ^3+534279040*A^2*B^3*C^4+267895296*A^2*B^2*C^5+54183680*A^2*B*C^6+215936*A^2*C^ 7+30848*A*B^8+9133440*A*B^7*C+54183680*A*B^6*C^2+135027328*A*B^5*C^3+179892480* A*B^4*C^4+135027328*A*B^3*C^5+54183680*A*B^2*C^6+9133440*A*B*C^7+30848*A*C^8+ 30848*B^8*C+215936*B^7*C^2+647808*B^6*C^3+1079680*B^5*C^4+1079680*B^4*C^5+ 647808*B^3*C^6+215936*B^2*C^7+30848*B*C^8-8960*A^7-62720*A^6*B-62720*A^6*C-\ 188160*A^5*B^2-376320*A^5*B*C-188160*A^5*C^2-313600*A^4*B^3-940800*A^4*B^2*C-\ 940800*A^4*B*C^2-313600*A^4*C^3-313600*A^3*B^4-1254400*A^3*B^3*C-1881600*A^3*B^ 2*C^2-1254400*A^3*B*C^3-313600*A^3*C^4-188160*A^2*B^5-940800*A^2*B^4*C-1881600* A^2*B^3*C^2-1881600*A^2*B^2*C^3-940800*A^2*B*C^4-188160*A^2*C^5-62720*A*B^6-\ 376320*A*B^5*C-940800*A*B^4*C^2-1254400*A*B^3*C^3-940800*A*B^2*C^4-376320*A*B*C ^5-62720*A*C^6-8960*B^7-62720*B^6*C-188160*B^5*C^2-313600*B^4*C^3-313600*B^3*C^ 4-188160*B^2*C^5-62720*B*C^6-8960*C^7)/(A^2*B+A^2*C+A*B^2-3*A*B*C+A*C^2+B^2*C+B *C^2-A-B-C)^4 Note that if N is large and the starting capitals are x1*N,x2*N,x3*N, (where\ , of course, x1+x2+x3=1), then the limit of the, 8, -th scaled moment about the mean tends to 20 20 19 2 19 19 2 (420 x1 x2 + 420 x1 x3 + 8820 x1 x2 + 17640 x1 x2 x3 + 8820 x1 x3 18 3 18 2 18 2 + 86818 x1 x2 + 245252 x1 x2 x3 + 245252 x1 x2 x3 18 3 17 4 17 3 + 86818 x1 x3 + 532342 x1 x2 + 1857114 x1 x2 x3 17 2 2 17 3 17 4 + 2649544 x1 x2 x3 + 1857114 x1 x2 x3 + 532342 x1 x3 16 5 16 4 16 3 2 + 2279198 x1 x2 + 9145920 x1 x2 x3 + 16316590 x1 x2 x3 16 2 3 16 4 16 5 + 16316590 x1 x2 x3 + 9145920 x1 x2 x3 + 2279198 x1 x3 15 6 15 5 15 4 2 + 7238362 x1 x2 + 32048506 x1 x2 x3 + 67120106 x1 x2 x3 15 3 3 15 2 4 15 5 + 84619924 x1 x2 x3 + 67120106 x1 x2 x3 + 32048506 x1 x2 x3 15 6 14 7 14 6 + 7238362 x1 x3 + 17680481 x1 x2 + 84361708 x1 x2 x3 14 5 2 14 4 3 14 3 4 + 201434100 x1 x2 x3 + 302943102 x1 x2 x3 + 302943102 x1 x2 x3 14 2 5 14 6 14 7 + 201434100 x1 x2 x3 + 84361708 x1 x2 x3 + 17680481 x1 x3 13 8 13 7 13 6 2 + 33985999 x1 x2 + 173015883 x1 x2 x3 + 465994544 x1 x2 x3 13 5 3 13 4 4 13 3 5 + 811783710 x1 x2 x3 + 969638100 x1 x2 x3 + 811783710 x1 x2 x3 13 2 6 13 7 13 8 + 465994544 x1 x2 x3 + 173015883 x1 x2 x3 + 33985999 x1 x3 12 9 12 8 12 7 2 + 52166413 x1 x2 + 283733610 x1 x2 x3 + 859409137 x1 x2 x3 12 6 3 12 5 4 + 1693963498 x1 x2 x3 + 2331701694 x1 x2 x3 12 4 5 12 3 6 + 2331701694 x1 x2 x3 + 1693963498 x1 x2 x3 12 2 7 12 8 12 9 + 859409137 x1 x2 x3 + 283733610 x1 x2 x3 + 52166413 x1 x3 11 10 11 9 11 8 2 + 64498395 x1 x2 + 378929993 x1 x2 x3 + 1285930477 x1 x2 x3 11 7 3 11 6 4 + 2808546949 x1 x2 x3 + 4371523832 x1 x2 x3 11 5 5 11 4 6 + 5068951524 x1 x2 x3 + 4371523832 x1 x2 x3 11 3 7 11 2 8 + 2808546949 x1 x2 x3 + 1285930477 x1 x2 x3 11 9 11 10 10 11 + 378929993 x1 x2 x3 + 64498395 x1 x3 + 64498395 x1 x2 10 10 10 9 2 + 416764452 x1 x2 x3 + 1571747606 x1 x2 x3 10 8 3 10 7 4 + 3762105516 x1 x2 x3 + 6631140633 x1 x2 x3 10 6 5 10 5 6 + 8812715236 x1 x2 x3 + 8812715236 x1 x2 x3 10 4 7 10 3 8 + 6631140633 x1 x2 x3 + 3762105516 x1 x2 x3 10 2 9 10 10 10 11 + 1571747606 x1 x2 x3 + 416764452 x1 x2 x3 + 64498395 x1 x3 9 12 9 11 9 10 2 + 52166413 x1 x2 + 378929993 x1 x2 x3 + 1571747606 x1 x2 x3 9 9 3 9 8 4 + 4137215654 x1 x2 x3 + 8218923427 x1 x2 x3 9 7 5 9 6 6 + 11921506883 x1 x2 x3 + 13189630168 x1 x2 x3 9 5 7 9 4 8 + 11921506883 x1 x2 x3 + 8218923427 x1 x2 x3 9 3 9 9 2 10 9 11 + 4137215654 x1 x2 x3 + 1571747606 x1 x2 x3 + 378929993 x1 x2 x3 9 12 8 13 8 12 + 52166413 x1 x3 + 33985999 x1 x2 + 283733610 x1 x2 x3 8 11 2 8 10 3 + 1285930477 x1 x2 x3 + 3762105516 x1 x2 x3 8 9 4 8 8 5 + 8218923427 x1 x2 x3 + 12972320790 x1 x2 x3 8 7 6 8 6 7 + 17360966337 x1 x2 x3 + 17360966337 x1 x2 x3 8 5 8 8 4 9 + 12972320790 x1 x2 x3 + 8218923427 x1 x2 x3 8 3 10 8 2 11 + 3762105516 x1 x2 x3 + 1285930477 x1 x2 x3 8 12 8 13 7 14 + 283733610 x1 x2 x3 + 33985999 x1 x3 + 17680481 x1 x2 7 13 7 12 2 7 11 3 + 173015883 x1 x2 x3 + 859409137 x1 x2 x3 + 2808546949 x1 x2 x3 7 10 4 7 9 5 + 6631140633 x1 x2 x3 + 11921506883 x1 x2 x3 7 8 6 7 7 7 + 17360966337 x1 x2 x3 + 17451600309 x1 x2 x3 7 6 8 7 5 9 + 17360966337 x1 x2 x3 + 11921506883 x1 x2 x3 7 4 10 7 3 11 + 6631140633 x1 x2 x3 + 2808546949 x1 x2 x3 7 2 12 7 13 7 14 + 859409137 x1 x2 x3 + 173015883 x1 x2 x3 + 17680481 x1 x3 6 15 6 14 6 13 2 + 7238362 x1 x2 + 84361708 x1 x2 x3 + 465994544 x1 x2 x3 6 12 3 6 11 4 + 1693963498 x1 x2 x3 + 4371523832 x1 x2 x3 6 10 5 6 9 6 + 8812715236 x1 x2 x3 + 13189630168 x1 x2 x3 6 8 7 6 7 8 + 17360966337 x1 x2 x3 + 17360966337 x1 x2 x3 6 6 9 6 5 10 + 13189630168 x1 x2 x3 + 8812715236 x1 x2 x3 6 4 11 6 3 12 + 4371523832 x1 x2 x3 + 1693963498 x1 x2 x3 6 2 13 6 14 6 15 + 465994544 x1 x2 x3 + 84361708 x1 x2 x3 + 7238362 x1 x3 5 16 5 15 5 14 2 + 2279198 x1 x2 + 32048506 x1 x2 x3 + 201434100 x1 x2 x3 5 13 3 5 12 4 + 811783710 x1 x2 x3 + 2331701694 x1 x2 x3 5 11 5 5 10 6 + 5068951524 x1 x2 x3 + 8812715236 x1 x2 x3 5 9 7 5 8 8 + 11921506883 x1 x2 x3 + 12972320790 x1 x2 x3 5 7 9 5 6 10 + 11921506883 x1 x2 x3 + 8812715236 x1 x2 x3 5 5 11 5 4 12 + 5068951524 x1 x2 x3 + 2331701694 x1 x2 x3 5 3 13 5 2 14 5 15 + 811783710 x1 x2 x3 + 201434100 x1 x2 x3 + 32048506 x1 x2 x3 5 16 4 17 4 16 + 2279198 x1 x3 + 532342 x1 x2 + 9145920 x1 x2 x3 4 15 2 4 14 3 4 13 4 + 67120106 x1 x2 x3 + 302943102 x1 x2 x3 + 969638100 x1 x2 x3 4 12 5 4 11 6 + 2331701694 x1 x2 x3 + 4371523832 x1 x2 x3 4 10 7 4 9 8 + 6631140633 x1 x2 x3 + 8218923427 x1 x2 x3 4 8 9 4 7 10 + 8218923427 x1 x2 x3 + 6631140633 x1 x2 x3 4 6 11 4 5 12 + 4371523832 x1 x2 x3 + 2331701694 x1 x2 x3 4 4 13 4 3 14 4 2 15 + 969638100 x1 x2 x3 + 302943102 x1 x2 x3 + 67120106 x1 x2 x3 4 16 4 17 3 18 + 9145920 x1 x2 x3 + 532342 x1 x3 + 86818 x1 x2 3 17 3 16 2 3 15 3 + 1857114 x1 x2 x3 + 16316590 x1 x2 x3 + 84619924 x1 x2 x3 3 14 4 3 13 5 + 302943102 x1 x2 x3 + 811783710 x1 x2 x3 3 12 6 3 11 7 + 1693963498 x1 x2 x3 + 2808546949 x1 x2 x3 3 10 8 3 9 9 + 3762105516 x1 x2 x3 + 4137215654 x1 x2 x3 3 8 10 3 7 11 + 3762105516 x1 x2 x3 + 2808546949 x1 x2 x3 3 6 12 3 5 13 + 1693963498 x1 x2 x3 + 811783710 x1 x2 x3 3 4 14 3 3 15 3 2 16 + 302943102 x1 x2 x3 + 84619924 x1 x2 x3 + 16316590 x1 x2 x3 3 17 3 18 2 19 + 1857114 x1 x2 x3 + 86818 x1 x3 + 8820 x1 x2 2 18 2 17 2 2 16 3 + 245252 x1 x2 x3 + 2649544 x1 x2 x3 + 16316590 x1 x2 x3 2 15 4 2 14 5 2 13 6 + 67120106 x1 x2 x3 + 201434100 x1 x2 x3 + 465994544 x1 x2 x3 2 12 7 2 11 8 + 859409137 x1 x2 x3 + 1285930477 x1 x2 x3 2 10 9 2 9 10 + 1571747606 x1 x2 x3 + 1571747606 x1 x2 x3 2 8 11 2 7 12 + 1285930477 x1 x2 x3 + 859409137 x1 x2 x3 2 6 13 2 5 14 2 4 15 + 465994544 x1 x2 x3 + 201434100 x1 x2 x3 + 67120106 x1 x2 x3 2 3 16 2 2 17 2 18 + 16316590 x1 x2 x3 + 2649544 x1 x2 x3 + 245252 x1 x2 x3 2 19 20 19 18 2 + 8820 x1 x3 + 420 x1 x2 + 17640 x1 x2 x3 + 245252 x1 x2 x3 17 3 16 4 15 5 + 1857114 x1 x2 x3 + 9145920 x1 x2 x3 + 32048506 x1 x2 x3 14 6 13 7 12 8 + 84361708 x1 x2 x3 + 173015883 x1 x2 x3 + 283733610 x1 x2 x3 11 9 10 10 9 11 + 378929993 x1 x2 x3 + 416764452 x1 x2 x3 + 378929993 x1 x2 x3 8 12 7 13 6 14 + 283733610 x1 x2 x3 + 173015883 x1 x2 x3 + 84361708 x1 x2 x3 5 15 4 16 3 17 + 32048506 x1 x2 x3 + 9145920 x1 x2 x3 + 1857114 x1 x2 x3 2 18 19 20 20 + 245252 x1 x2 x3 + 17640 x1 x2 x3 + 420 x1 x3 + 420 x2 x3 19 2 18 3 17 4 16 5 + 8820 x2 x3 + 86818 x2 x3 + 532342 x2 x3 + 2279198 x2 x3 15 6 14 7 13 8 + 7238362 x2 x3 + 17680481 x2 x3 + 33985999 x2 x3 12 9 11 10 10 11 + 52166413 x2 x3 + 64498395 x2 x3 + 64498395 x2 x3 9 12 8 13 7 14 + 52166413 x2 x3 + 33985999 x2 x3 + 17680481 x2 x3 6 15 5 16 4 17 3 18 + 7238362 x2 x3 + 2279198 x2 x3 + 532342 x2 x3 + 86818 x2 x3 2 19 20 / 3 3 3 + 8820 x2 x3 + 420 x2 x3 ) / (600600 x1 x2 x3 / 2 2 2 2 2 2 4 (x1 x2 + x1 x3 + x1 x2 - 3 x1 x2 x3 + x1 x3 + x2 x3 + x2 x3 ) ) and in Maple notation 1/600600*(420*x1^20*x2+420*x1^20*x3+8820*x1^19*x2^2+17640*x1^19*x2*x3+8820*x1^ 19*x3^2+86818*x1^18*x2^3+245252*x1^18*x2^2*x3+245252*x1^18*x2*x3^2+86818*x1^18* x3^3+532342*x1^17*x2^4+1857114*x1^17*x2^3*x3+2649544*x1^17*x2^2*x3^2+1857114*x1 ^17*x2*x3^3+532342*x1^17*x3^4+2279198*x1^16*x2^5+9145920*x1^16*x2^4*x3+16316590 *x1^16*x2^3*x3^2+16316590*x1^16*x2^2*x3^3+9145920*x1^16*x2*x3^4+2279198*x1^16* x3^5+7238362*x1^15*x2^6+32048506*x1^15*x2^5*x3+67120106*x1^15*x2^4*x3^2+ 84619924*x1^15*x2^3*x3^3+67120106*x1^15*x2^2*x3^4+32048506*x1^15*x2*x3^5+ 7238362*x1^15*x3^6+17680481*x1^14*x2^7+84361708*x1^14*x2^6*x3+201434100*x1^14* x2^5*x3^2+302943102*x1^14*x2^4*x3^3+302943102*x1^14*x2^3*x3^4+201434100*x1^14* x2^2*x3^5+84361708*x1^14*x2*x3^6+17680481*x1^14*x3^7+33985999*x1^13*x2^8+ 173015883*x1^13*x2^7*x3+465994544*x1^13*x2^6*x3^2+811783710*x1^13*x2^5*x3^3+ 969638100*x1^13*x2^4*x3^4+811783710*x1^13*x2^3*x3^5+465994544*x1^13*x2^2*x3^6+ 173015883*x1^13*x2*x3^7+33985999*x1^13*x3^8+52166413*x1^12*x2^9+283733610*x1^12 *x2^8*x3+859409137*x1^12*x2^7*x3^2+1693963498*x1^12*x2^6*x3^3+2331701694*x1^12* x2^5*x3^4+2331701694*x1^12*x2^4*x3^5+1693963498*x1^12*x2^3*x3^6+859409137*x1^12 *x2^2*x3^7+283733610*x1^12*x2*x3^8+52166413*x1^12*x3^9+64498395*x1^11*x2^10+ 378929993*x1^11*x2^9*x3+1285930477*x1^11*x2^8*x3^2+2808546949*x1^11*x2^7*x3^3+ 4371523832*x1^11*x2^6*x3^4+5068951524*x1^11*x2^5*x3^5+4371523832*x1^11*x2^4*x3^ 6+2808546949*x1^11*x2^3*x3^7+1285930477*x1^11*x2^2*x3^8+378929993*x1^11*x2*x3^9 +64498395*x1^11*x3^10+64498395*x1^10*x2^11+416764452*x1^10*x2^10*x3+1571747606* x1^10*x2^9*x3^2+3762105516*x1^10*x2^8*x3^3+6631140633*x1^10*x2^7*x3^4+ 8812715236*x1^10*x2^6*x3^5+8812715236*x1^10*x2^5*x3^6+6631140633*x1^10*x2^4*x3^ 7+3762105516*x1^10*x2^3*x3^8+1571747606*x1^10*x2^2*x3^9+416764452*x1^10*x2*x3^ 10+64498395*x1^10*x3^11+52166413*x1^9*x2^12+378929993*x1^9*x2^11*x3+1571747606* x1^9*x2^10*x3^2+4137215654*x1^9*x2^9*x3^3+8218923427*x1^9*x2^8*x3^4+11921506883 *x1^9*x2^7*x3^5+13189630168*x1^9*x2^6*x3^6+11921506883*x1^9*x2^5*x3^7+ 8218923427*x1^9*x2^4*x3^8+4137215654*x1^9*x2^3*x3^9+1571747606*x1^9*x2^2*x3^10+ 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10+1285930477*x1^2*x2^8*x3^11+859409137*x1^2*x2^7*x3^12+465994544*x1^2*x2^6*x3^ 13+201434100*x1^2*x2^5*x3^14+67120106*x1^2*x2^4*x3^15+16316590*x1^2*x2^3*x3^16+ 2649544*x1^2*x2^2*x3^17+245252*x1^2*x2*x3^18+8820*x1^2*x3^19+420*x1*x2^20+17640 *x1*x2^19*x3+245252*x1*x2^18*x3^2+1857114*x1*x2^17*x3^3+9145920*x1*x2^16*x3^4+ 32048506*x1*x2^15*x3^5+84361708*x1*x2^14*x3^6+173015883*x1*x2^13*x3^7+283733610 *x1*x2^12*x3^8+378929993*x1*x2^11*x3^9+416764452*x1*x2^10*x3^10+378929993*x1*x2 ^9*x3^11+283733610*x1*x2^8*x3^12+173015883*x1*x2^7*x3^13+84361708*x1*x2^6*x3^14 +32048506*x1*x2^5*x3^15+9145920*x1*x2^4*x3^16+1857114*x1*x2^3*x3^17+245252*x1* x2^2*x3^18+17640*x1*x2*x3^19+420*x1*x3^20+420*x2^20*x3+8820*x2^19*x3^2+86818*x2 ^18*x3^3+532342*x2^17*x3^4+2279198*x2^16*x3^5+7238362*x2^15*x3^6+17680481*x2^14 *x3^7+33985999*x2^13*x3^8+52166413*x2^12*x3^9+64498395*x2^11*x3^10+64498395*x2^ 10*x3^11+52166413*x2^9*x3^12+33985999*x2^8*x3^13+17680481*x2^7*x3^14+7238362*x2 ^6*x3^15+2279198*x2^5*x3^16+532342*x2^4*x3^17+86818*x2^3*x3^18+8820*x2^2*x3^19+ 420*x2*x3^20)/x1^3/x2^3/x3^3/(x1^2*x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2* x3+x2*x3^2)^4 The , 9, -th (straight) moment of the Duration is: 15 15 14 2 14 3 A B C (878931 A B + 878931 A C + 14062896 A B + 28125792 A B C 14 2 13 3 13 2 + 14062896 A C + 102580020 A B + 310631760 A B C 13 2 13 3 12 4 + 310631760 A B C + 102580020 A C + 451717560 A B 12 3 12 2 2 12 3 + 1847354040 A B C + 2791272960 A B C + 1847354040 A B C 12 4 11 5 11 4 + 451717560 A C + 1340315748 A B + 6957111384 A B C 11 3 2 11 2 3 11 4 + 14173561440 A B C + 14173561440 A B C + 6957111384 A B C 11 5 10 6 10 5 + 1340315748 A C + 2832264144 A B + 17954818992 A B C 10 4 2 10 3 3 10 2 4 + 46374571008 A B C + 62504032320 A B C + 46374571008 A B C 10 5 10 6 9 7 + 17954818992 A B C + 2832264144 A C + 4392400446 A B 9 6 9 5 2 9 4 3 + 33147054576 A B C + 104483913984 A B C + 178465957344 A B C 9 3 4 9 2 5 9 6 + 178465957344 A B C + 104483913984 A B C + 33147054576 A B C 9 7 8 8 8 7 + 4392400446 A C + 5076261900 A B + 44805180780 A B C 8 6 2 8 5 3 8 4 4 + 168053667840 A B C + 350215086240 A B C + 443780674560 A B C 8 3 5 8 2 6 8 7 + 350215086240 A B C + 168053667840 A B C + 44805180780 A B C 8 8 7 9 7 8 + 5076261900 A C + 4392400446 A B + 44805180780 A B C 7 7 2 7 6 3 7 5 4 + 196530935040 A B C + 487269082080 A B C + 753258413952 A B C 7 4 5 7 3 6 7 2 7 + 753258413952 A B C + 487269082080 A B C + 196530935040 A B C 7 8 7 9 6 10 + 44805180780 A B C + 4392400446 A C + 2832264144 A B 6 9 6 8 2 6 7 3 + 33147054576 A B C + 168053667840 A B C + 487269082080 A B C 6 6 4 6 5 5 6 4 6 + 896149039104 A B C + 1093237668864 A B C + 896149039104 A B C 6 3 7 6 2 8 6 9 + 487269082080 A B C + 168053667840 A B C + 33147054576 A B C 6 10 5 11 5 10 + 2832264144 A C + 1340315748 A B + 17954818992 A B C 5 9 2 5 8 3 5 7 4 + 104483913984 A B C + 350215086240 A B C + 753258413952 A B C 5 6 5 5 5 6 5 4 7 + 1093237668864 A B C + 1093237668864 A B C + 753258413952 A B C 5 3 8 5 2 9 5 10 + 350215086240 A B C + 104483913984 A B C + 17954818992 A B C 5 11 4 12 4 11 + 1340315748 A C + 451717560 A B + 6957111384 A B C 4 10 2 4 9 3 4 8 4 + 46374571008 A B C + 178465957344 A B C + 443780674560 A B C 4 7 5 4 6 6 4 5 7 + 753258413952 A B C + 896149039104 A B C + 753258413952 A B C 4 4 8 4 3 9 4 2 10 + 443780674560 A B C + 178465957344 A B C + 46374571008 A B C 4 11 4 12 3 13 + 6957111384 A B C + 451717560 A C + 102580020 A B 3 12 3 11 2 3 10 3 + 1847354040 A B C + 14173561440 A B C + 62504032320 A B C 3 9 4 3 8 5 3 7 6 + 178465957344 A B C + 350215086240 A B C + 487269082080 A B C 3 6 7 3 5 8 3 4 9 + 487269082080 A B C + 350215086240 A B C + 178465957344 A B C 3 3 10 3 2 11 3 12 + 62504032320 A B C + 14173561440 A B C + 1847354040 A B C 3 13 2 14 2 13 + 102580020 A C + 14062896 A B + 310631760 A B C 2 12 2 2 11 3 2 10 4 + 2791272960 A B C + 14173561440 A B C + 46374571008 A B C 2 9 5 2 8 6 2 7 7 + 104483913984 A B C + 168053667840 A B C + 196530935040 A B C 2 6 8 2 5 9 2 4 10 + 168053667840 A B C + 104483913984 A B C + 46374571008 A B C 2 3 11 2 2 12 2 13 + 14173561440 A B C + 2791272960 A B C + 310631760 A B C 2 14 15 14 13 2 + 14062896 A C + 878931 A B + 28125792 A B C + 310631760 A B C 12 3 11 4 10 5 + 1847354040 A B C + 6957111384 A B C + 17954818992 A B C 9 6 8 7 7 8 + 33147054576 A B C + 44805180780 A B C + 44805180780 A B C 6 9 5 10 4 11 + 33147054576 A B C + 17954818992 A B C + 6957111384 A B C 3 12 2 13 14 + 1847354040 A B C + 310631760 A B C + 28125792 A B C 15 15 14 2 13 3 + 878931 A C + 878931 B C + 14062896 B C + 102580020 B C 12 4 11 5 10 6 + 451717560 B C + 1340315748 B C + 2832264144 B C 9 7 8 8 7 9 + 4392400446 B C + 5076261900 B C + 4392400446 B C 6 10 5 11 4 12 + 2832264144 B C + 1340315748 B C + 451717560 B C 3 13 2 14 15 13 + 102580020 B C + 14062896 B C + 878931 B C - 23133600 A B 13 12 2 12 12 2 - 23133600 A C - 323870400 A B - 647740800 A B C - 323870400 A C 11 3 11 2 11 2 - 2029037040 A B - 6163231680 A B C - 6163231680 A B C 11 3 10 4 10 3 - 2029037040 A C - 7507183680 A B - 30942181440 A B C 10 2 2 10 3 10 4 - 46869995520 A B C - 30942181440 A B C - 7507183680 A C 9 5 9 4 9 3 2 - 18233022240 A B - 95988576960 A B C - 196900865280 A B C 9 2 3 9 4 9 5 - 196900865280 A B C - 95988576960 A B C - 18233022240 A C 8 6 8 5 8 4 2 - 30569400000 A B - 198158011200 A B C - 518509900800 A B C 8 3 3 8 2 4 8 5 - 701842579200 A B C - 518509900800 A B C - 198158011200 A B C 8 6 7 7 7 6 - 30569400000 A C - 36230687640 A B - 282485050560 A B C 7 5 2 7 4 3 7 3 4 - 909470384640 A B C - 1569671602560 A B C - 1569671602560 A B C 7 2 5 7 6 7 7 - 909470384640 A B C - 282485050560 A B C - 36230687640 A C 6 8 6 7 6 6 2 - 30569400000 A B - 282485050560 A B C - 1093661429760 A B C 6 5 3 6 4 4 6 3 5 - 2321957767680 A B C - 2960423976960 A B C - 2321957767680 A B C 6 2 6 6 7 6 8 - 1093661429760 A B C - 282485050560 A B C - 30569400000 A C 5 9 5 8 5 7 2 - 18233022240 A B - 198158011200 A B C - 909470384640 A B C 5 6 3 5 5 4 5 4 5 - 2321957767680 A B C - 3641561763840 A B C - 3641561763840 A B C 5 3 6 5 2 7 5 8 - 2321957767680 A B C - 909470384640 A B C - 198158011200 A B C 5 9 4 10 4 9 - 18233022240 A C - 7507183680 A B - 95988576960 A B C 4 8 2 4 7 3 4 6 4 - 518509900800 A B C - 1569671602560 A B C - 2960423976960 A B C 4 5 5 4 4 6 4 3 7 - 3641561763840 A B C - 2960423976960 A B C - 1569671602560 A B C 4 2 8 4 9 4 10 - 518509900800 A B C - 95988576960 A B C - 7507183680 A C 3 11 3 10 3 9 2 - 2029037040 A B - 30942181440 A B C - 196900865280 A B C 3 8 3 3 7 4 3 6 5 - 701842579200 A B C - 1569671602560 A B C - 2321957767680 A B C 3 5 6 3 4 7 3 3 8 - 2321957767680 A B C - 1569671602560 A B C - 701842579200 A B C 3 2 9 3 10 3 11 - 196900865280 A B C - 30942181440 A B C - 2029037040 A C 2 12 2 11 2 10 2 - 323870400 A B - 6163231680 A B C - 46869995520 A B C 2 9 3 2 8 4 2 7 5 - 196900865280 A B C - 518509900800 A B C - 909470384640 A B C 2 6 6 2 5 7 2 4 8 - 1093661429760 A B C - 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323870400*A^2*C^12-23133600*A*B^13-647740800*A*B^12*C-6163231680*A*B^11*C^2-\ 30942181440*A*B^10*C^3-95988576960*A*B^9*C^4-198158011200*A*B^8*C^5-\ 282485050560*A*B^7*C^6-282485050560*A*B^6*C^7-198158011200*A*B^5*C^8-\ 95988576960*A*B^4*C^9-30942181440*A*B^3*C^10-6163231680*A*B^2*C^11-647740800*A* B*C^12-23133600*A*C^13-23133600*B^13*C-323870400*B^12*C^2-2029037040*B^11*C^3-\ 7507183680*B^10*C^4-18233022240*B^9*C^5-30569400000*B^8*C^6-36230687640*B^7*C^7 -30569400000*B^6*C^8-18233022240*B^5*C^9-7507183680*B^4*C^10-2029037040*B^3*C^ 11-323870400*B^2*C^12-23133600*B*C^13+217577934*A^11*B+217577934*A^11*C+ 2610935208*A^10*B^2+5221870416*A^10*B*C+2610935208*A^10*C^2+13643805294*A^9*B^3 +41647754232*A^9*B^2*C+41647754232*A^9*B*C^2+13643805294*A^9*C^3+40703761980*A^ 8*B^4+169978431420*A^8*B^3*C+258549338880*A^8*B^2*C^2+169978431420*A^8*B*C^3+ 40703761980*A^8*C^4+76411418202*A^7*B^5+412401183516*A^7*B^4*C+856092026160*A^7 *B^3*C^2+856092026160*A^7*B^2*C^3+412401183516*A^7*B*C^4+76411418202*A^7*C^5+ 93925654704*A^6*B^6+634385464272*A^6*B^5*C+1699775497728*A^6*B^4*C^2+ 2318631376320*A^6*B^3*C^3+1699775497728*A^6*B^2*C^4+634385464272*A^6*B*C^5+ 93925654704*A^6*C^6+76411418202*A^5*B^7+634385464272*A^5*B^6*C+2126391982848*A^ 5*B^5*C^2+3742803959328*A^5*B^4*C^3+3742803959328*A^5*B^3*C^4+2126391982848*A^5 *B^2*C^5+634385464272*A^5*B*C^6+76411418202*A^5*C^7+40703761980*A^4*B^8+ 412401183516*A^4*B^7*C+1699775497728*A^4*B^6*C^2+3742803959328*A^4*B^5*C^3+ 4829451766272*A^4*B^4*C^4+3742803959328*A^4*B^3*C^5+1699775497728*A^4*B^2*C^6+ 412401183516*A^4*B*C^7+40703761980*A^4*C^8+13643805294*A^3*B^9+169978431420*A^3 *B^8*C+856092026160*A^3*B^7*C^2+2318631376320*A^3*B^6*C^3+3742803959328*A^3*B^5 *C^4+3742803959328*A^3*B^4*C^5+2318631376320*A^3*B^3*C^6+856092026160*A^3*B^2*C ^7+169978431420*A^3*B*C^8+13643805294*A^3*C^9+2610935208*A^2*B^10+41647754232*A ^2*B^9*C+258549338880*A^2*B^8*C^2+856092026160*A^2*B^7*C^3+1699775497728*A^2*B^ 6*C^4+2126391982848*A^2*B^5*C^5+1699775497728*A^2*B^4*C^6+856092026160*A^2*B^3* C^7+258549338880*A^2*B^2*C^8+41647754232*A^2*B*C^9+2610935208*A^2*C^10+ 217577934*A*B^11+5221870416*A*B^10*C+41647754232*A*B^9*C^2+169978431420*A*B^8*C ^3+412401183516*A*B^7*C^4+634385464272*A*B^6*C^5+634385464272*A*B^5*C^6+ 412401183516*A*B^4*C^7+169978431420*A*B^3*C^8+41647754232*A*B^2*C^9+5221870416* A*B*C^10+217577934*A*C^11+217577934*B^11*C+2610935208*B^10*C^2+13643805294*B^9* C^3+40703761980*B^8*C^4+76411418202*B^7*C^5+93925654704*B^6*C^6+76411418202*B^5 *C^7+40703761980*B^4*C^8+13643805294*B^3*C^9+2610935208*B^2*C^10+217577934*B*C^ 11-848507400*A^9*B-848507400*A^9*C-8485074000*A^8*B^2-16970148000*A^8*B*C-\ 8485074000*A^8*C^2-35382758580*A^7*B^3-108948350160*A^7*B^2*C-108948350160*A^7* B*C^2-35382758580*A^7*C^3-79420292640*A^6*B^4-340081765920*A^6*B^3*C-\ 521322946560*A^6*B^2*C^2-340081765920*A^6*B*C^3-79420292640*A^6*C^4-\ 103517902800*A^5*B^5-588524732640*A^5*B^4*C-1251718116480*A^5*B^3*C^2-\ 1251718116480*A^5*B^2*C^3-588524732640*A^5*B*C^4-103517902800*A^5*C^5-\ 79420292640*A^4*B^6-588524732640*A^4*B^5*C-1661716892160*A^4*B^4*C^2-\ 2305224904320*A^4*B^3*C^3-1661716892160*A^4*B^2*C^4-588524732640*A^4*B*C^5-\ 79420292640*A^4*C^6-35382758580*A^3*B^7-340081765920*A^3*B^6*C-1251718116480*A^ 3*B^5*C^2-2305224904320*A^3*B^4*C^3-2305224904320*A^3*B^3*C^4-1251718116480*A^3 *B^2*C^5-340081765920*A^3*B*C^6-35382758580*A^3*C^7-8485074000*A^2*B^8-\ 108948350160*A^2*B^7*C-521322946560*A^2*B^6*C^2-1251718116480*A^2*B^5*C^3-\ 1661716892160*A^2*B^4*C^4-1251718116480*A^2*B^3*C^5-521322946560*A^2*B^2*C^6-\ 108948350160*A^2*B*C^7-8485074000*A^2*C^8-848507400*A*B^9-16970148000*A*B^8*C-\ 108948350160*A*B^7*C^2-340081765920*A*B^6*C^3-588524732640*A*B^5*C^4-\ 588524732640*A*B^4*C^5-340081765920*A*B^3*C^6-108948350160*A*B^2*C^7-\ 16970148000*A*B*C^8-848507400*A*C^9-848507400*B^9*C-8485074000*B^8*C^2-\ 35382758580*B^7*C^3-79420292640*B^6*C^4-103517902800*B^5*C^5-79420292640*B^4*C^ 6-35382758580*B^3*C^7-8485074000*B^2*C^8-848507400*B*C^9+1118126295*A^7*B+ 1118126295*A^7*C+8945010360*A^6*B^2+17890020720*A^6*B*C+8945010360*A^6*C^2+ 27580448610*A^5*B^3+86468433480*A^5*B^2*C+86468433480*A^5*B*C^2+27580448610*A^5 *C^3+40252546620*A^4*B^4+183372712380*A^4*B^3*C+286240331520*A^4*B^2*C^2+ 183372712380*A^4*B*C^3+40252546620*A^4*C^4+27580448610*A^3*B^5+183372712380*A^3 *B^4*C+417433816800*A^3*B^3*C^2+417433816800*A^3*B^2*C^3+183372712380*A^3*B*C^4 +27580448610*A^3*C^5+8945010360*A^2*B^6+86468433480*A^2*B^5*C+286240331520*A^2* B^4*C^2+417433816800*A^2*B^3*C^3+286240331520*A^2*B^2*C^4+86468433480*A^2*B*C^5 +8945010360*A^2*C^6+1118126295*A*B^7+17890020720*A*B^6*C+86468433480*A*B^5*C^2+ 183372712380*A*B^4*C^3+183372712380*A*B^3*C^4+86468433480*A*B^2*C^5+17890020720 *A*B*C^6+1118126295*A*C^7+1118126295*B^7*C+8945010360*B^6*C^2+27580448610*B^5*C ^3+40252546620*B^4*C^4+27580448610*B^3*C^5+8945010360*B^2*C^6+1118126295*B*C^7-\ 91334880*A^5*B-91334880*A^5*C-548009280*A^4*B^2-1096018560*A^4*B*C-548009280*A^ 4*C^2-1050351120*A^3*B^3-3470725440*A^3*B^2*C-3470725440*A^3*B*C^2-1050351120*A ^3*C^3-548009280*A^2*B^4-3470725440*A^2*B^3*C-5845432320*A^2*B^2*C^2-3470725440 *A^2*B*C^3-548009280*A^2*C^4-91334880*A*B^5-1096018560*A*B^4*C-3470725440*A*B^3 *C^2-3470725440*A*B^2*C^3-1096018560*A*B*C^4-91334880*A*C^5-91334880*B^5*C-\ 548009280*B^4*C^2-1050351120*B^3*C^3-548009280*B^2*C^4-91334880*B*C^5+184901520 *A^3*B+184901520*A^3*C+739606080*A^2*B^2+1479212160*A^2*B*C+739606080*A^2*C^2+ 184901520*A*B^3+1479212160*A*B^2*C+1479212160*A*B*C^2+184901520*A*C^3+184901520 *B^3*C+739606080*B^2*C^2+184901520*B*C^3+910199040*A*B+910199040*A*C+910199040* B*C+567750400)/(A+B+C) Hence the, 9, -th moment about the mean is 23 23 22 2 22 3 A B C (878931 A B + 878931 A C + 21094344 A B + 42188688 A B C 22 2 21 3 21 2 + 21094344 A C + 239693256 A B + 687271068 A B C 21 2 21 3 20 4 + 687271068 A B C + 239693256 A C + 1715338944 A B 20 3 20 2 2 20 3 + 6196264776 A B C + 8961851664 A B C + 6196264776 A B C 20 4 19 5 19 4 + 1715338944 A C + 8675344134 A B + 36864188154 A B C 19 3 2 19 2 3 19 4 + 67790554020 A B C + 67790554020 A B C + 36864188154 A B C 19 5 18 6 18 5 + 8675344134 A C + 32980985784 A B + 158204795544 A B C 18 4 2 18 3 3 18 2 4 + 346228525440 A B C + 442009431360 A B C + 346228525440 A B C 18 5 18 6 17 7 + 158204795544 A B C + 32980985784 A C + 97868271546 A B 17 6 17 5 2 + 516485112372 A B C + 1296322253316 A B C 17 4 3 17 3 4 + 1984704837660 A B C + 1984704837660 A B C 17 2 5 17 6 17 7 + 1296322253316 A B C + 516485112372 A B C + 97868271546 A C 16 8 16 7 16 6 2 + 232342046244 A B + 1328320183572 A B C + 3744670207752 A B C 16 5 3 16 4 4 + 6639520844664 A B C + 7981657548480 A B C 16 3 5 16 2 6 + 6639520844664 A B C + 3744670207752 A B C 16 7 16 8 15 9 + 1328320183572 A B C + 232342046244 A C + 448700408577 A B 15 8 15 7 2 + 2757648095739 A B C + 8632395357876 A B C 15 6 3 15 5 4 + 17343385518924 A B C + 24195830790618 A B C 15 4 5 15 3 6 + 24195830790618 A B C + 17343385518924 A B C 15 2 7 15 8 15 9 + 8632395357876 A B C + 2757648095739 A B C + 448700408577 A C 14 10 14 9 14 8 2 + 712880192592 A B + 4702521090492 A B C + 16241616819900 A B C 14 7 3 14 6 4 + 36410172649920 A B C + 57609355592088 A B C 14 5 5 14 4 6 + 66902317728336 A B C + 57609355592088 A B C 14 3 7 14 2 8 + 36410172649920 A B C + 16241616819900 A B C 14 9 14 10 13 11 + 4702521090492 A B C + 712880192592 A C + 938576292516 A B 13 10 13 9 2 + 6668070157476 A B C + 25302557810520 A B C 13 8 3 13 7 4 + 62573896235880 A B C + 110661554431620 A B C 13 6 5 13 5 6 + 146090355865272 A B C + 146090355865272 A B C 13 4 7 13 3 8 + 110661554431620 A B C + 62573896235880 A B C 13 2 9 13 10 + 25302557810520 A B C + 6668070157476 A B C 13 11 12 12 12 11 + 938576292516 A C + 1028242462104 A B + 7924752656388 A B C 12 10 2 12 9 3 + 32918513931684 A B C + 89137422490860 A B C 12 8 4 12 7 5 + 174711543608340 A B C + 258338086219368 A B C 12 6 6 12 5 7 + 293483922728976 A B C + 258338086219368 A B C 12 4 8 12 3 9 + 174711543608340 A B C + 89137422490860 A B C 12 2 10 12 11 + 32918513931684 A B C + 7924752656388 A B C 12 12 11 13 11 12 + 1028242462104 A C + 938576292516 A B + 7924752656388 A B C 11 11 2 11 10 3 + 35920518367968 A B C + 106147489098456 A B C 11 9 4 11 8 5 + 229179399238764 A B C + 374385773578752 A B C 11 7 6 11 6 7 + 474867823305096 A B C + 474867823305096 A B C 11 5 8 11 4 9 + 374385773578752 A B C + 229179399238764 A B C 11 3 10 11 2 11 + 106147489098456 A B C + 35920518367968 A B C 11 12 11 13 10 14 + 7924752656388 A B C + 938576292516 A C + 712880192592 A B 10 13 10 12 2 + 6668070157476 A B C + 32918513931684 A B C 10 11 3 10 10 4 + 106147489098456 A B C + 250815705747048 A B C 10 9 5 10 8 6 + 448739072671500 A B C + 633712321672092 A B C 10 7 7 10 6 8 + 713209579231968 A B C + 633712321672092 A B C 10 5 9 10 4 10 + 448739072671500 A B C + 250815705747048 A B C 10 3 11 10 2 12 + 106147489098456 A B C + 32918513931684 A B C 10 13 10 14 9 15 + 6668070157476 A B C + 712880192592 A C + 448700408577 A B 9 14 9 13 2 + 4702521090492 A B C + 25302557810520 A B C 9 12 3 9 11 4 + 89137422490860 A B C + 229179399238764 A B C 9 10 5 9 9 6 + 448739072671500 A B C + 699530402770056 A B C 9 8 7 9 7 8 + 858402276384510 A B C + 858402276384510 A B C 9 6 9 9 5 10 + 699530402770056 A B C + 448739072671500 A B C 9 4 11 9 3 12 + 229179399238764 A B C + 89137422490860 A B C 9 2 13 9 14 9 15 + 25302557810520 A B C + 4702521090492 A B C + 448700408577 A C 8 16 8 15 8 14 2 + 232342046244 A B + 2757648095739 A B C + 16241616819900 A B C 8 13 3 8 12 4 + 62573896235880 A B C + 174711543608340 A B C 8 11 5 8 10 6 + 374385773578752 A B C + 633712321672092 A B C 8 9 7 8 8 8 + 858402276384510 A B C + 969724206954360 A B C 8 7 9 8 6 10 + 858402276384510 A B C + 633712321672092 A B C 8 5 11 8 4 12 + 374385773578752 A B C + 174711543608340 A B C 8 3 13 8 2 14 + 62573896235880 A B C + 16241616819900 A B C 8 15 8 16 7 17 + 2757648095739 A B C + 232342046244 A C + 97868271546 A B 7 16 7 15 2 + 1328320183572 A B C + 8632395357876 A B C 7 14 3 7 13 4 + 36410172649920 A B C + 110661554431620 A B C 7 12 5 7 11 6 + 258338086219368 A B C + 474867823305096 A B C 7 10 7 7 9 8 + 713209579231968 A B C + 858402276384510 A B C 7 8 9 7 7 10 + 858402276384510 A B C + 713209579231968 A B C 7 6 11 7 5 12 + 474867823305096 A B C + 258338086219368 A B C 7 4 13 7 3 14 + 110661554431620 A B C + 36410172649920 A B C 7 2 15 7 16 7 17 + 8632395357876 A B C + 1328320183572 A B C + 97868271546 A C 6 18 6 17 6 16 2 + 32980985784 A B + 516485112372 A B C + 3744670207752 A B C 6 15 3 6 14 4 + 17343385518924 A B C + 57609355592088 A B C 6 13 5 6 12 6 + 146090355865272 A B C + 293483922728976 A B C 6 11 7 6 10 8 + 474867823305096 A B C + 633712321672092 A B C 6 9 9 6 8 10 + 699530402770056 A B C + 633712321672092 A B C 6 7 11 6 6 12 + 474867823305096 A B C + 293483922728976 A B C 6 5 13 6 4 14 + 146090355865272 A B C + 57609355592088 A B C 6 3 15 6 2 16 + 17343385518924 A B C + 3744670207752 A B C 6 17 6 18 5 19 + 516485112372 A B C + 32980985784 A C + 8675344134 A B 5 18 5 17 2 + 158204795544 A B C + 1296322253316 A B C 5 16 3 5 15 4 + 6639520844664 A B C + 24195830790618 A B C 5 14 5 5 13 6 + 66902317728336 A B C + 146090355865272 A B C 5 12 7 5 11 8 + 258338086219368 A B C + 374385773578752 A B C 5 10 9 5 9 10 + 448739072671500 A B C + 448739072671500 A B C 5 8 11 5 7 12 + 374385773578752 A B C + 258338086219368 A B C 5 6 13 5 5 14 + 146090355865272 A B C + 66902317728336 A B C 5 4 15 5 3 16 + 24195830790618 A B C + 6639520844664 A B C 5 2 17 5 18 5 19 + 1296322253316 A B C + 158204795544 A B C + 8675344134 A C 4 20 4 19 4 18 2 + 1715338944 A B + 36864188154 A B C + 346228525440 A B C 4 17 3 4 16 4 + 1984704837660 A B C + 7981657548480 A B C 4 15 5 4 14 6 + 24195830790618 A B C + 57609355592088 A B C 4 13 7 4 12 8 + 110661554431620 A B C + 174711543608340 A B C 4 11 9 4 10 10 + 229179399238764 A B C + 250815705747048 A B C 4 9 11 4 8 12 + 229179399238764 A B C + 174711543608340 A B C 4 7 13 4 6 14 + 110661554431620 A B C + 57609355592088 A B C 4 5 15 4 4 16 + 24195830790618 A B C + 7981657548480 A B C 4 3 17 4 2 18 4 19 + 1984704837660 A B C + 346228525440 A B C + 36864188154 A B C 4 20 3 21 3 20 + 1715338944 A C + 239693256 A B + 6196264776 A B C 3 19 2 3 18 3 3 17 4 + 67790554020 A B C + 442009431360 A B C + 1984704837660 A B C 3 16 5 3 15 6 + 6639520844664 A B C + 17343385518924 A B C 3 14 7 3 13 8 + 36410172649920 A B C + 62573896235880 A B C 3 12 9 3 11 10 + 89137422490860 A B C + 106147489098456 A B C 3 10 11 3 9 12 + 106147489098456 A B C + 89137422490860 A B C 3 8 13 3 7 14 + 62573896235880 A B C + 36410172649920 A B C 3 6 15 3 5 16 + 17343385518924 A B C + 6639520844664 A B C 3 4 17 3 3 18 3 2 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35598859690680*A*B^6*C^9+19782555532740*A*B^5*C^10+7934386750290*A*B^4*C^11+ 2191095646740*A*B^3*C^12+382996708380*A*B^2*C^13+35780041440*A*B*C^14+ 1118126295*A*C^15+1118126295*B^15*C+17890020720*B^14*C^2+130448067750*B^13*C^3+ 573971498100*B^12*C^4+1701042803460*B^11*C^5+3589930824480*B^10*C^6+ 5561932900095*B^9*C^7+6425499108600*B^8*C^8+5561932900095*B^7*C^9+3589930824480 *B^6*C^10+1701042803460*B^5*C^11+573971498100*B^4*C^12+130448067750*B^3*C^13+ 17890020720*B^2*C^14+1118126295*B*C^15-91334880*A^13*B-91334880*A^13*C-\ 1278688320*A^12*B^2-2557376640*A^12*B*C-1278688320*A^12*C^2-7991802000*A^11*B^3 -25498025280*A^11*B^2*C-25498025280*A^11*B*C^2-7991802000*A^11*C^3-29409831360* A^10*B^4-134707809600*A^10*B^3*C-210595956480*A^10*B^2*C^2-134707809600*A^10*B* C^3-29409831360*A^10*C^4-70967201760*A^9*B^5-434803037760*A^9*B^4*C-\ 1011562755840*A^9*B^3*C^2-1011562755840*A^9*B^2*C^3-434803037760*A^9*B*C^4-\ 70967201760*A^9*C^5-118370004480*A^8*B^6-920702371680*A^8*B^5*C-2998096395840*A ^8*B^4*C^2-4321092157200*A^8*B^3*C^3-2998096395840*A^8*B^2*C^4-920702371680*A^8 *B*C^5-118370004480*A^8*C^6-140016371040*A^7*B^7-1328508155520*A^7*B^6*C-\ 5666438232000*A^7*B^5*C^2-11045267382240*A^7*B^4*C^3-11045267382240*A^7*B^3*C^4 -5666438232000*A^7*B^2*C^5-1328508155520*A^7*B*C^6-140016371040*A^7*C^7-\ 118370004480*A^6*B^8-1328508155520*A^6*B^7*C-6987055944960*A^6*B^6*C^2-\ 17486820187200*A^6*B^5*C^3-23419804786560*A^6*B^4*C^4-17486820187200*A^6*B^3*C^ 5-6987055944960*A^6*B^2*C^6-1328508155520*A^6*B*C^7-118370004480*A^6*C^8-\ 70967201760*A^5*B^9-920702371680*A^5*B^8*C-5666438232000*A^5*B^7*C^2-\ 17486820187200*A^5*B^6*C^3-29934481187520*A^5*B^5*C^4-29934481187520*A^5*B^4*C^ 5-17486820187200*A^5*B^3*C^6-5666438232000*A^5*B^2*C^7-920702371680*A^5*B*C^8-\ 70967201760*A^5*C^9-29409831360*A^4*B^10-434803037760*A^4*B^9*C-2998096395840*A ^4*B^8*C^2-11045267382240*A^4*B^7*C^3-23419804786560*A^4*B^6*C^4-29934481187520 *A^4*B^5*C^5-23419804786560*A^4*B^4*C^6-11045267382240*A^4*B^3*C^7-\ 2998096395840*A^4*B^2*C^8-434803037760*A^4*B*C^9-29409831360*A^4*C^10-\ 7991802000*A^3*B^11-134707809600*A^3*B^10*C-1011562755840*A^3*B^9*C^2-\ 4321092157200*A^3*B^8*C^3-11045267382240*A^3*B^7*C^4-17486820187200*A^3*B^6*C^5 -17486820187200*A^3*B^5*C^6-11045267382240*A^3*B^4*C^7-4321092157200*A^3*B^3*C^ 8-1011562755840*A^3*B^2*C^9-134707809600*A^3*B*C^10-7991802000*A^3*C^11-\ 1278688320*A^2*B^12-25498025280*A^2*B^11*C-210595956480*A^2*B^10*C^2-\ 1011562755840*A^2*B^9*C^3-2998096395840*A^2*B^8*C^4-5666438232000*A^2*B^7*C^5-\ 6987055944960*A^2*B^6*C^6-5666438232000*A^2*B^5*C^7-2998096395840*A^2*B^4*C^8-\ 1011562755840*A^2*B^3*C^9-210595956480*A^2*B^2*C^10-25498025280*A^2*B*C^11-\ 1278688320*A^2*C^12-91334880*A*B^13-2557376640*A*B^12*C-25498025280*A*B^11*C^2-\ 134707809600*A*B^10*C^3-434803037760*A*B^9*C^4-920702371680*A*B^8*C^5-\ 1328508155520*A*B^7*C^6-1328508155520*A*B^6*C^7-920702371680*A*B^5*C^8-\ 434803037760*A*B^4*C^9-134707809600*A*B^3*C^10-25498025280*A*B^2*C^11-\ 2557376640*A*B*C^12-91334880*A*C^13-91334880*B^13*C-1278688320*B^12*C^2-\ 7991802000*B^11*C^3-29409831360*B^10*C^4-70967201760*B^9*C^5-118370004480*B^8*C ^6-140016371040*B^7*C^7-118370004480*B^6*C^8-70967201760*B^5*C^9-29409831360*B^ 4*C^10-7991802000*B^3*C^11-1278688320*B^2*C^12-91334880*B*C^13+184901520*A^11*B +184901520*A^11*C+2218818240*A^10*B^2+4437636480*A^10*B*C+2218818240*A^10*C^2+ 11278992720*A^9*B^3+34478301120*A^9*B^2*C+34478301120*A^9*B*C^2+11278992720*A^9 *C^3+32542667520*A^8*B^4+135769339920*A^8*B^3*C+166741358400*A^8*B^2*C^2+ 135769339920*A^8*B*C^3+32542667520*A^8*C^4+59538289440*A^7*B^5+319109994720*A^7 *B^4*C+421823040000*A^7*B^3*C^2+421823040000*A^7*B^2*C^3+319109994720*A^7*B*C^4 +59538289440*A^7*C^5+72481395840*A^6*B^6+481926580800*A^6*B^5*C+682802296320*A^ 6*B^4*C^2+546714222720*A^6*B^3*C^3+682802296320*A^6*B^2*C^4+481926580800*A^6*B* C^5+72481395840*A^6*C^6+59538289440*A^5*B^7+481926580800*A^5*B^6*C+790922263680 *A^5*B^5*C^2+379901913120*A^5*B^4*C^3+379901913120*A^5*B^3*C^4+790922263680*A^5 *B^2*C^5+481926580800*A^5*B*C^6+59538289440*A^5*C^7+32542667520*A^4*B^8+ 319109994720*A^4*B^7*C+682802296320*A^4*B^6*C^2+379901913120*A^4*B^5*C^3-\ 32666112000*A^4*B^4*C^4+379901913120*A^4*B^3*C^5+682802296320*A^4*B^2*C^6+ 319109994720*A^4*B*C^7+32542667520*A^4*C^8+11278992720*A^3*B^9+135769339920*A^3 *B^8*C+421823040000*A^3*B^7*C^2+546714222720*A^3*B^6*C^3+379901913120*A^3*B^5*C ^4+379901913120*A^3*B^4*C^5+546714222720*A^3*B^3*C^6+421823040000*A^3*B^2*C^7+ 135769339920*A^3*B*C^8+11278992720*A^3*C^9+2218818240*A^2*B^10+34478301120*A^2* B^9*C+166741358400*A^2*B^8*C^2+421823040000*A^2*B^7*C^3+682802296320*A^2*B^6*C^ 4+790922263680*A^2*B^5*C^5+682802296320*A^2*B^4*C^6+421823040000*A^2*B^3*C^7+ 166741358400*A^2*B^2*C^8+34478301120*A^2*B*C^9+2218818240*A^2*C^10+184901520*A* B^11+4437636480*A*B^10*C+34478301120*A*B^9*C^2+135769339920*A*B^8*C^3+ 319109994720*A*B^7*C^4+481926580800*A*B^6*C^5+481926580800*A*B^5*C^6+ 319109994720*A*B^4*C^7+135769339920*A*B^3*C^8+34478301120*A*B^2*C^9+4437636480* A*B*C^10+184901520*A*C^11+184901520*B^11*C+2218818240*B^10*C^2+11278992720*B^9* C^3+32542667520*B^8*C^4+59538289440*B^7*C^5+72481395840*B^6*C^6+59538289440*B^5 *C^7+32542667520*B^4*C^8+11278992720*B^3*C^9+2218818240*B^2*C^10+184901520*B*C^ 11+910199040*A^9*B+910199040*A^9*C+7281592320*A^8*B^2+15555636480*A^8*B*C+ 7281592320*A^8*C^2+25485573120*A^7*B^3+84314081280*A^7*B^2*C+84314081280*A^7*B* C^2+25485573120*A^7*C^3+50971146240*A^6*B^4+231097466880*A^6*B^3*C+360252641280 *A^6*B^2*C^2+231097466880*A^6*B*C^3+50971146240*A^6*C^4+63713932800*A^5*B^5+ 372419658240*A^5*B^4*C+798689310720*A^5*B^3*C^2+798689310720*A^5*B^2*C^3+ 372419658240*A^5*B*C^4+63713932800*A^5*C^5+50971146240*A^4*B^6+372419658240*A^4 *B^5*C+1030938316800*A^4*B^4*C^2+1418979609600*A^4*B^3*C^3+1030938316800*A^4*B^ 2*C^4+372419658240*A^4*B*C^5+50971146240*A^4*C^6+25485573120*A^3*B^7+ 231097466880*A^3*B^6*C+798689310720*A^3*B^5*C^2+1418979609600*A^3*B^4*C^3+ 1418979609600*A^3*B^3*C^4+798689310720*A^3*B^2*C^5+231097466880*A^3*B*C^6+ 25485573120*A^3*C^7+7281592320*A^2*B^8+84314081280*A^2*B^7*C+360252641280*A^2*B ^6*C^2+798689310720*A^2*B^5*C^3+1030938316800*A^2*B^4*C^4+798689310720*A^2*B^3* C^5+360252641280*A^2*B^2*C^6+84314081280*A^2*B*C^7+7281592320*A^2*C^8+910199040 *A*B^9+15555636480*A*B^8*C+84314081280*A*B^7*C^2+231097466880*A*B^6*C^3+ 372419658240*A*B^5*C^4+372419658240*A*B^4*C^5+231097466880*A*B^3*C^6+ 84314081280*A*B^2*C^7+15555636480*A*B*C^8+910199040*A*C^9+910199040*B^9*C+ 7281592320*B^8*C^2+25485573120*B^7*C^3+50971146240*B^6*C^4+63713932800*B^5*C^5+ 50971146240*B^4*C^6+25485573120*B^3*C^7+7281592320*B^2*C^8+910199040*B*C^9+ 567750400*A^8+4542003200*A^7*B+4542003200*A^7*C+15897011200*A^6*B^2+31794022400 *A^6*B*C+15897011200*A^6*C^2+31794022400*A^5*B^3+95382067200*A^5*B^2*C+ 95382067200*A^5*B*C^2+31794022400*A^5*C^3+39742528000*A^4*B^4+158970112000*A^4* B^3*C+238455168000*A^4*B^2*C^2+158970112000*A^4*B*C^3+39742528000*A^4*C^4+ 31794022400*A^3*B^5+158970112000*A^3*B^4*C+317940224000*A^3*B^3*C^2+ 317940224000*A^3*B^2*C^3+158970112000*A^3*B*C^4+31794022400*A^3*C^5+15897011200 *A^2*B^6+95382067200*A^2*B^5*C+238455168000*A^2*B^4*C^2+317940224000*A^2*B^3*C^ 3+238455168000*A^2*B^2*C^4+95382067200*A^2*B*C^5+15897011200*A^2*C^6+4542003200 *A*B^7+31794022400*A*B^6*C+95382067200*A*B^5*C^2+158970112000*A*B^4*C^3+ 158970112000*A*B^3*C^4+95382067200*A*B^2*C^5+31794022400*A*B*C^6+4542003200*A*C ^7+567750400*B^8+4542003200*B^7*C+15897011200*B^6*C^2+31794022400*B^5*C^3+ 39742528000*B^4*C^4+31794022400*B^3*C^5+15897011200*B^2*C^6+4542003200*B*C^7+ 567750400*C^8)/(A+B+C)^9 Hence the scaled, 9, -th moment about the mean is 23 23 22 2 22 A B C (878931 A B + 878931 A C + 21094344 A B + 42188688 A B C 22 2 21 3 21 2 + 21094344 A C + 239693256 A B + 687271068 A B C 21 2 21 3 20 4 + 687271068 A B C + 239693256 A C + 1715338944 A B 20 3 20 2 2 20 3 + 6196264776 A B C + 8961851664 A B C + 6196264776 A B C 20 4 19 5 19 4 + 1715338944 A C + 8675344134 A B + 36864188154 A B C 19 3 2 19 2 3 19 4 + 67790554020 A B C + 67790554020 A B C + 36864188154 A B C 19 5 18 6 18 5 + 8675344134 A C + 32980985784 A B + 158204795544 A B C 18 4 2 18 3 3 18 2 4 + 346228525440 A B C + 442009431360 A B C + 346228525440 A B C 18 5 18 6 17 7 + 158204795544 A B C + 32980985784 A C + 97868271546 A B 17 6 17 5 2 + 516485112372 A B C + 1296322253316 A B C 17 4 3 17 3 4 + 1984704837660 A B C + 1984704837660 A B C 17 2 5 17 6 17 7 + 1296322253316 A B C + 516485112372 A B C + 97868271546 A C 16 8 16 7 16 6 2 + 232342046244 A B + 1328320183572 A B C + 3744670207752 A B C 16 5 3 16 4 4 + 6639520844664 A B C + 7981657548480 A B C 16 3 5 16 2 6 + 6639520844664 A B C + 3744670207752 A B C 16 7 16 8 15 9 + 1328320183572 A B C + 232342046244 A C + 448700408577 A B 15 8 15 7 2 + 2757648095739 A B C + 8632395357876 A B C 15 6 3 15 5 4 + 17343385518924 A B C + 24195830790618 A B C 15 4 5 15 3 6 + 24195830790618 A B C + 17343385518924 A B C 15 2 7 15 8 15 9 + 8632395357876 A B C + 2757648095739 A B C + 448700408577 A C 14 10 14 9 14 8 2 + 712880192592 A B + 4702521090492 A B C + 16241616819900 A B C 14 7 3 14 6 4 + 36410172649920 A B C + 57609355592088 A B C 14 5 5 14 4 6 + 66902317728336 A B C + 57609355592088 A B C 14 3 7 14 2 8 + 36410172649920 A B C + 16241616819900 A B C 14 9 14 10 13 11 + 4702521090492 A B C + 712880192592 A C + 938576292516 A B 13 10 13 9 2 + 6668070157476 A B C + 25302557810520 A B C 13 8 3 13 7 4 + 62573896235880 A B C + 110661554431620 A B C 13 6 5 13 5 6 + 146090355865272 A B C + 146090355865272 A B C 13 4 7 13 3 8 + 110661554431620 A B C + 62573896235880 A B C 13 2 9 13 10 + 25302557810520 A B C + 6668070157476 A B C 13 11 12 12 12 11 + 938576292516 A C + 1028242462104 A B + 7924752656388 A B C 12 10 2 12 9 3 + 32918513931684 A B C + 89137422490860 A B C 12 8 4 12 7 5 + 174711543608340 A B C + 258338086219368 A B C 12 6 6 12 5 7 + 293483922728976 A B C + 258338086219368 A B C 12 4 8 12 3 9 + 174711543608340 A B C + 89137422490860 A B C 12 2 10 12 11 + 32918513931684 A B C + 7924752656388 A B C 12 12 11 13 11 12 + 1028242462104 A C + 938576292516 A B + 7924752656388 A B C 11 11 2 11 10 3 + 35920518367968 A B C + 106147489098456 A B C 11 9 4 11 8 5 + 229179399238764 A B C + 374385773578752 A B C 11 7 6 11 6 7 + 474867823305096 A B C + 474867823305096 A B C 11 5 8 11 4 9 + 374385773578752 A B C + 229179399238764 A B C 11 3 10 11 2 11 + 106147489098456 A B C + 35920518367968 A B C 11 12 11 13 10 14 + 7924752656388 A B C + 938576292516 A C + 712880192592 A B 10 13 10 12 2 + 6668070157476 A B C + 32918513931684 A B C 10 11 3 10 10 4 + 106147489098456 A B C + 250815705747048 A B C 10 9 5 10 8 6 + 448739072671500 A B C + 633712321672092 A B C 10 7 7 10 6 8 + 713209579231968 A B C + 633712321672092 A B C 10 5 9 10 4 10 + 448739072671500 A B C + 250815705747048 A B C 10 3 11 10 2 12 + 106147489098456 A B 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100415353750920*A*B^7*C^10-61313726864520*A*B^6*C^11-28547468665560*A*B^5*C^12-\ 9811557433080*A*B^4*C^13-2363978134440*A*B^3*C^14-365788317960*A*B^2*C^15-\ 30546266400*A*B*C^16-848507400*A*C^17-848507400*B^17*C-15273133200*B^16*C^2-\ 127021557780*B^15*C^3-647580847680*B^14*C^4-2264157146160*B^13*C^5-\ 5754237783840*B^12*C^6-10992498217740*B^11*C^7-16097542989840*B^10*C^8-\ 18260218650960*B^9*C^9-16097542989840*B^8*C^10-10992498217740*B^7*C^11-\ 5754237783840*B^6*C^12-2264157146160*B^5*C^13-647580847680*B^4*C^14-\ 127021557780*B^3*C^15-15273133200*B^2*C^16-848507400*B*C^17+1118126295*A^15*B+ 1118126295*A^15*C+17890020720*A^14*B^2+35780041440*A^14*B*C+17890020720*A^14*C^ 2+130448067750*A^13*B^3+382996708380*A^13*B^2*C+382996708380*A^13*B*C^2+ 130448067750*A^13*C^3+573971498100*A^12*B^4+2191095646740*A^12*B^3*C+ 3234248297280*A^12*B^2*C^2+2191095646740*A^12*B*C^3+573971498100*A^12*C^4+ 1701042803460*A^11*B^5+7934386750290*A^11*B^4*C+15277443641460*A^11*B^3*C^2+ 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*B*C^8+5561932900095*A^7*C^9+3589930824480*A^6*B^10+35598859690680*A^6*B^9*C+ 153075392123940*A^6*B^8*C^2+387773058172500*A^6*B^7*C^3+657336008609280*A^6*B^6 *C^4+781258827389040*A^6*B^5*C^5+657336008609280*A^6*B^4*C^6+387773058172500*A^ 6*B^3*C^7+153075392123940*A^6*B^2*C^8+35598859690680*A^6*B*C^9+3589930824480*A^ 6*C^10+1701042803460*A^5*B^11+19782555532740*A^5*B^10*C+98959293292860*A^5*B^9* C^2+289329582652110*A^5*B^8*C^3+564841072185390*A^5*B^7*C^4+781258827389040*A^5 *B^6*C^5+781258827389040*A^5*B^5*C^6+564841072185390*A^5*B^4*C^7+ 289329582652110*A^5*B^3*C^8+98959293292860*A^5*B^2*C^9+19782555532740*A^5*B*C^ 10+1701042803460*A^5*C^11+573971498100*A^4*B^12+7934386750290*A^4*B^11*C+ 46583993596140*A^4*B^10*C^2+158120817734880*A^4*B^9*C^3+355070307271680*A^4*B^8 *C^4+564841072185390*A^4*B^7*C^5+657336008609280*A^4*B^6*C^6+564841072185390*A^ 4*B^5*C^7+355070307271680*A^4*B^4*C^8+158120817734880*A^4*B^3*C^9+ 46583993596140*A^4*B^2*C^10+7934386750290*A^4*B*C^11+573971498100*A^4*C^12+ 130448067750*A^3*B^13+2191095646740*A^3*B^12*C+15277443641460*A^3*B^11*C^2+ 60782737775760*A^3*B^10*C^3+158120817734880*A^3*B^9*C^4+289329582652110*A^3*B^8 *C^5+387773058172500*A^3*B^7*C^6+387773058172500*A^3*B^6*C^7+289329582652110*A^ 3*B^5*C^8+158120817734880*A^3*B^4*C^9+60782737775760*A^3*B^3*C^10+ 15277443641460*A^3*B^2*C^11+2191095646740*A^3*B*C^12+130448067750*A^3*C^13+ 17890020720*A^2*B^14+382996708380*A^2*B^13*C+3234248297280*A^2*B^12*C^2+ 15277443641460*A^2*B^11*C^3+46583993596140*A^2*B^10*C^4+98959293292860*A^2*B^9* C^5+153075392123940*A^2*B^8*C^6+176583580790760*A^2*B^7*C^7+153075392123940*A^2 *B^6*C^8+98959293292860*A^2*B^5*C^9+46583993596140*A^2*B^4*C^10+15277443641460* A^2*B^3*C^11+3234248297280*A^2*B^2*C^12+382996708380*A^2*B*C^13+17890020720*A^2 *C^14+1118126295*A*B^15+35780041440*A*B^14*C+382996708380*A*B^13*C^2+ 2191095646740*A*B^12*C^3+7934386750290*A*B^11*C^4+19782555532740*A*B^10*C^5+ 35598859690680*A*B^9*C^6+47469242987595*A*B^8*C^7+47469242987595*A*B^7*C^8+ 35598859690680*A*B^6*C^9+19782555532740*A*B^5*C^10+7934386750290*A*B^4*C^11+ 2191095646740*A*B^3*C^12+382996708380*A*B^2*C^13+35780041440*A*B*C^14+ 1118126295*A*C^15+1118126295*B^15*C+17890020720*B^14*C^2+130448067750*B^13*C^3+ 573971498100*B^12*C^4+1701042803460*B^11*C^5+3589930824480*B^10*C^6+ 5561932900095*B^9*C^7+6425499108600*B^8*C^8+5561932900095*B^7*C^9+3589930824480 *B^6*C^10+1701042803460*B^5*C^11+573971498100*B^4*C^12+130448067750*B^3*C^13+ 17890020720*B^2*C^14+1118126295*B*C^15-91334880*A^13*B-91334880*A^13*C-\ 1278688320*A^12*B^2-2557376640*A^12*B*C-1278688320*A^12*C^2-7991802000*A^11*B^3 -25498025280*A^11*B^2*C-25498025280*A^11*B*C^2-7991802000*A^11*C^3-29409831360* A^10*B^4-134707809600*A^10*B^3*C-210595956480*A^10*B^2*C^2-134707809600*A^10*B* C^3-29409831360*A^10*C^4-70967201760*A^9*B^5-434803037760*A^9*B^4*C-\ 1011562755840*A^9*B^3*C^2-1011562755840*A^9*B^2*C^3-434803037760*A^9*B*C^4-\ 70967201760*A^9*C^5-118370004480*A^8*B^6-920702371680*A^8*B^5*C-2998096395840*A ^8*B^4*C^2-4321092157200*A^8*B^3*C^3-2998096395840*A^8*B^2*C^4-920702371680*A^8 *B*C^5-118370004480*A^8*C^6-140016371040*A^7*B^7-1328508155520*A^7*B^6*C-\ 5666438232000*A^7*B^5*C^2-11045267382240*A^7*B^4*C^3-11045267382240*A^7*B^3*C^4 -5666438232000*A^7*B^2*C^5-1328508155520*A^7*B*C^6-140016371040*A^7*C^7-\ 118370004480*A^6*B^8-1328508155520*A^6*B^7*C-6987055944960*A^6*B^6*C^2-\ 17486820187200*A^6*B^5*C^3-23419804786560*A^6*B^4*C^4-17486820187200*A^6*B^3*C^ 5-6987055944960*A^6*B^2*C^6-1328508155520*A^6*B*C^7-118370004480*A^6*C^8-\ 70967201760*A^5*B^9-920702371680*A^5*B^8*C-5666438232000*A^5*B^7*C^2-\ 17486820187200*A^5*B^6*C^3-29934481187520*A^5*B^5*C^4-29934481187520*A^5*B^4*C^ 5-17486820187200*A^5*B^3*C^6-5666438232000*A^5*B^2*C^7-920702371680*A^5*B*C^8-\ 70967201760*A^5*C^9-29409831360*A^4*B^10-434803037760*A^4*B^9*C-2998096395840*A ^4*B^8*C^2-11045267382240*A^4*B^7*C^3-23419804786560*A^4*B^6*C^4-29934481187520 *A^4*B^5*C^5-23419804786560*A^4*B^4*C^6-11045267382240*A^4*B^3*C^7-\ 2998096395840*A^4*B^2*C^8-434803037760*A^4*B*C^9-29409831360*A^4*C^10-\ 7991802000*A^3*B^11-134707809600*A^3*B^10*C-1011562755840*A^3*B^9*C^2-\ 4321092157200*A^3*B^8*C^3-11045267382240*A^3*B^7*C^4-17486820187200*A^3*B^6*C^5 -17486820187200*A^3*B^5*C^6-11045267382240*A^3*B^4*C^7-4321092157200*A^3*B^3*C^ 8-1011562755840*A^3*B^2*C^9-134707809600*A^3*B*C^10-7991802000*A^3*C^11-\ 1278688320*A^2*B^12-25498025280*A^2*B^11*C-210595956480*A^2*B^10*C^2-\ 1011562755840*A^2*B^9*C^3-2998096395840*A^2*B^8*C^4-5666438232000*A^2*B^7*C^5-\ 6987055944960*A^2*B^6*C^6-5666438232000*A^2*B^5*C^7-2998096395840*A^2*B^4*C^8-\ 1011562755840*A^2*B^3*C^9-210595956480*A^2*B^2*C^10-25498025280*A^2*B*C^11-\ 1278688320*A^2*C^12-91334880*A*B^13-2557376640*A*B^12*C-25498025280*A*B^11*C^2-\ 134707809600*A*B^10*C^3-434803037760*A*B^9*C^4-920702371680*A*B^8*C^5-\ 1328508155520*A*B^7*C^6-1328508155520*A*B^6*C^7-920702371680*A*B^5*C^8-\ 434803037760*A*B^4*C^9-134707809600*A*B^3*C^10-25498025280*A*B^2*C^11-\ 2557376640*A*B*C^12-91334880*A*C^13-91334880*B^13*C-1278688320*B^12*C^2-\ 7991802000*B^11*C^3-29409831360*B^10*C^4-70967201760*B^9*C^5-118370004480*B^8*C ^6-140016371040*B^7*C^7-118370004480*B^6*C^8-70967201760*B^5*C^9-29409831360*B^ 4*C^10-7991802000*B^3*C^11-1278688320*B^2*C^12-91334880*B*C^13+184901520*A^11*B +184901520*A^11*C+2218818240*A^10*B^2+4437636480*A^10*B*C+2218818240*A^10*C^2+ 11278992720*A^9*B^3+34478301120*A^9*B^2*C+34478301120*A^9*B*C^2+11278992720*A^9 *C^3+32542667520*A^8*B^4+135769339920*A^8*B^3*C+166741358400*A^8*B^2*C^2+ 135769339920*A^8*B*C^3+32542667520*A^8*C^4+59538289440*A^7*B^5+319109994720*A^7 *B^4*C+421823040000*A^7*B^3*C^2+421823040000*A^7*B^2*C^3+319109994720*A^7*B*C^4 +59538289440*A^7*C^5+72481395840*A^6*B^6+481926580800*A^6*B^5*C+682802296320*A^ 6*B^4*C^2+546714222720*A^6*B^3*C^3+682802296320*A^6*B^2*C^4+481926580800*A^6*B* C^5+72481395840*A^6*C^6+59538289440*A^5*B^7+481926580800*A^5*B^6*C+790922263680 *A^5*B^5*C^2+379901913120*A^5*B^4*C^3+379901913120*A^5*B^3*C^4+790922263680*A^5 *B^2*C^5+481926580800*A^5*B*C^6+59538289440*A^5*C^7+32542667520*A^4*B^8+ 319109994720*A^4*B^7*C+682802296320*A^4*B^6*C^2+379901913120*A^4*B^5*C^3-\ 32666112000*A^4*B^4*C^4+379901913120*A^4*B^3*C^5+682802296320*A^4*B^2*C^6+ 319109994720*A^4*B*C^7+32542667520*A^4*C^8+11278992720*A^3*B^9+135769339920*A^3 *B^8*C+421823040000*A^3*B^7*C^2+546714222720*A^3*B^6*C^3+379901913120*A^3*B^5*C ^4+379901913120*A^3*B^4*C^5+546714222720*A^3*B^3*C^6+421823040000*A^3*B^2*C^7+ 135769339920*A^3*B*C^8+11278992720*A^3*C^9+2218818240*A^2*B^10+34478301120*A^2* B^9*C+166741358400*A^2*B^8*C^2+421823040000*A^2*B^7*C^3+682802296320*A^2*B^6*C^ 4+790922263680*A^2*B^5*C^5+682802296320*A^2*B^4*C^6+421823040000*A^2*B^3*C^7+ 166741358400*A^2*B^2*C^8+34478301120*A^2*B*C^9+2218818240*A^2*C^10+184901520*A* B^11+4437636480*A*B^10*C+34478301120*A*B^9*C^2+135769339920*A*B^8*C^3+ 319109994720*A*B^7*C^4+481926580800*A*B^6*C^5+481926580800*A*B^5*C^6+ 319109994720*A*B^4*C^7+135769339920*A*B^3*C^8+34478301120*A*B^2*C^9+4437636480* A*B*C^10+184901520*A*C^11+184901520*B^11*C+2218818240*B^10*C^2+11278992720*B^9* C^3+32542667520*B^8*C^4+59538289440*B^7*C^5+72481395840*B^6*C^6+59538289440*B^5 *C^7+32542667520*B^4*C^8+11278992720*B^3*C^9+2218818240*B^2*C^10+184901520*B*C^ 11+910199040*A^9*B+910199040*A^9*C+7281592320*A^8*B^2+15555636480*A^8*B*C+ 7281592320*A^8*C^2+25485573120*A^7*B^3+84314081280*A^7*B^2*C+84314081280*A^7*B* C^2+25485573120*A^7*C^3+50971146240*A^6*B^4+231097466880*A^6*B^3*C+360252641280 *A^6*B^2*C^2+231097466880*A^6*B*C^3+50971146240*A^6*C^4+63713932800*A^5*B^5+ 372419658240*A^5*B^4*C+798689310720*A^5*B^3*C^2+798689310720*A^5*B^2*C^3+ 372419658240*A^5*B*C^4+63713932800*A^5*C^5+50971146240*A^4*B^6+372419658240*A^4 *B^5*C+1030938316800*A^4*B^4*C^2+1418979609600*A^4*B^3*C^3+1030938316800*A^4*B^ 2*C^4+372419658240*A^4*B*C^5+50971146240*A^4*C^6+25485573120*A^3*B^7+ 231097466880*A^3*B^6*C+798689310720*A^3*B^5*C^2+1418979609600*A^3*B^4*C^3+ 1418979609600*A^3*B^3*C^4+798689310720*A^3*B^2*C^5+231097466880*A^3*B*C^6+ 25485573120*A^3*C^7+7281592320*A^2*B^8+84314081280*A^2*B^7*C+360252641280*A^2*B ^6*C^2+798689310720*A^2*B^5*C^3+1030938316800*A^2*B^4*C^4+798689310720*A^2*B^3* C^5+360252641280*A^2*B^2*C^6+84314081280*A^2*B*C^7+7281592320*A^2*C^8+910199040 *A*B^9+15555636480*A*B^8*C+84314081280*A*B^7*C^2+231097466880*A*B^6*C^3+ 372419658240*A*B^5*C^4+372419658240*A*B^4*C^5+231097466880*A*B^3*C^6+ 84314081280*A*B^2*C^7+15555636480*A*B*C^8+910199040*A*C^9+910199040*B^9*C+ 7281592320*B^8*C^2+25485573120*B^7*C^3+50971146240*B^6*C^4+63713932800*B^5*C^5+ 50971146240*B^4*C^6+25485573120*B^3*C^7+7281592320*B^2*C^8+910199040*B*C^9+ 567750400*A^8+4542003200*A^7*B+4542003200*A^7*C+15897011200*A^6*B^2+31794022400 *A^6*B*C+15897011200*A^6*C^2+31794022400*A^5*B^3+95382067200*A^5*B^2*C+ 95382067200*A^5*B*C^2+31794022400*A^5*C^3+39742528000*A^4*B^4+158970112000*A^4* B^3*C+238455168000*A^4*B^2*C^2+158970112000*A^4*B*C^3+39742528000*A^4*C^4+ 31794022400*A^3*B^5+158970112000*A^3*B^4*C+317940224000*A^3*B^3*C^2+ 317940224000*A^3*B^2*C^3+158970112000*A^3*B*C^4+31794022400*A^3*C^5+15897011200 *A^2*B^6+95382067200*A^2*B^5*C+238455168000*A^2*B^4*C^2+317940224000*A^2*B^3*C^ 3+238455168000*A^2*B^2*C^4+95382067200*A^2*B*C^5+15897011200*A^2*C^6+4542003200 *A*B^7+31794022400*A*B^6*C+95382067200*A*B^5*C^2+158970112000*A*B^4*C^3+ 158970112000*A*B^3*C^4+95382067200*A*B^2*C^5+31794022400*A*B*C^6+4542003200*A*C ^7+567750400*B^8+4542003200*B^7*C+15897011200*B^6*C^2+31794022400*B^5*C^3+ 39742528000*B^4*C^4+31794022400*B^3*C^5+15897011200*B^2*C^6+4542003200*B*C^7+ 567750400*C^8)/(A+B+C)^9*3^(1/2)*2^(1/2)/(A*B*C*(A^2*B+A^2*C+A*B^2-3*A*B*C+A*C^ 2+B^2*C+B*C^2-A-B-C)/(A+B+C)^2)^(9/2) Note that if N is large and the starting capitals are x1*N,x2*N,x3*N, (where\ , of course, x1+x2+x3=1), then the limit of the, 9, -th scaled moment about the mean tends to 6 14 7 1/2 1/2 6 13 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3 6 2 7 8 + 5717712000 %1 x2 x3 + 2450448000 %1 x2 x3 + 612612000 %1 x2 x3 9 9 9 + 68068000 %1 x1 + 68068000 %1 x2 + 68068000 %1 x3 7 6 2 + 4900896000 %1 x1 x2 x3 + 17153136000 %1 x1 x2 x3 6 2 5 3 + 17153136000 %1 x1 x2 x3 + 34306272000 %1 x1 x2 x3 5 2 2 5 3 + 51459408000 %1 x1 x2 x3 + 34306272000 %1 x1 x2 x3 4 4 4 3 2 + 42882840000 %1 x1 x2 x3 + 85765680000 %1 x1 x2 x3 4 2 3 4 4 + 85765680000 %1 x1 x2 x3 + 42882840000 %1 x1 x2 x3 3 5 3 4 2 + 34306272000 %1 x1 x2 x3 + 85765680000 %1 x1 x2 x3 3 3 3 3 2 4 + 114354240000 %1 x1 x2 x3 + 85765680000 %1 x1 x2 x3 3 5 2 6 + 34306272000 %1 x1 x2 x3 + 17153136000 %1 x1 x2 x3 2 5 2 2 4 3 + 51459408000 %1 x1 x2 x3 + 85765680000 %1 x1 x2 x3 2 3 4 2 2 5 + 85765680000 %1 x1 x2 x3 + 51459408000 %1 x1 x2 x3 2 6 7 + 17153136000 %1 x1 x2 x3 + 4900896000 %1 x1 x2 x3 6 2 5 3 + 17153136000 %1 x1 x2 x3 + 34306272000 %1 x1 x2 x3 4 4 3 5 + 42882840000 %1 x1 x2 x3 + 34306272000 %1 x1 x2 x3 2 6 7 + 17153136000 %1 x1 x2 x3 + 4900896000 %1 x1 x2 x3 ) / %1 := |x1 x2 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x3+x2*x3^2)/(x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2*x3+x3^2))^(9/2)*x1^3*x3^6+ 2450448000*(x1*x2*x3*(x1^2*x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2*x3+x2*x3^ 2)/(x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2*x3+x3^2))^(9/2)*x1^2*x2^7+2450448000*(x1*x2* x3*(x1^2*x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2*x3+x2*x3^2)/(x1^2+2*x1*x2+2 *x1*x3+x2^2+2*x2*x3+x3^2))^(9/2)*x1^2*x3^7+612612000*(x1*x2*x3*(x1^2*x2+x1^2*x3 +x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2*x3+x2*x3^2)/(x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2*x3 +x3^2))^(9/2)*x1*x2^8+612612000*(x1*x2*x3*(x1^2*x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+ x1*x3^2+x2^2*x3+x2*x3^2)/(x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2*x3+x3^2))^(9/2)*x1*x3^ 8+612612000*(x1*x2*x3*(x1^2*x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2*x3+x2*x3 ^2)/(x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2*x3+x3^2))^(9/2)*x2^8*x3+2450448000*(x1*x2* x3*(x1^2*x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2*x3+x2*x3^2)/(x1^2+2*x1*x2+2 *x1*x3+x2^2+2*x2*x3+x3^2))^(9/2)*x2^7*x3^2+5717712000*(x1*x2*x3*(x1^2*x2+x1^2* x3+x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2*x3+x2*x3^2)/(x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2* x3+x3^2))^(9/2)*x2^6*x3^3+8576568000*(x1*x2*x3*(x1^2*x2+x1^2*x3+x1*x2^2-3*x1*x2 *x3+x1*x3^2+x2^2*x3+x2*x3^2)/(x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2*x3+x3^2))^(9/2)*x2 ^5*x3^4+8576568000*(x1*x2*x3*(x1^2*x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2* x3+x2*x3^2)/(x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2*x3+x3^2))^(9/2)*x2^4*x3^5+ 5717712000*(x1*x2*x3*(x1^2*x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2*x3+x2*x3^ 2)/(x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2*x3+x3^2))^(9/2)*x2^3*x3^6+2450448000*(x1*x2* x3*(x1^2*x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2*x3+x2*x3^2)/(x1^2+2*x1*x2+2 *x1*x3+x2^2+2*x2*x3+x3^2))^(9/2)*x2^2*x3^7+612612000*(x1*x2*x3*(x1^2*x2+x1^2*x3 +x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2*x3+x2*x3^2)/(x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2*x3 +x3^2))^(9/2)*x2*x3^8+68068000*(x1*x2*x3*(x1^2*x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+x1 *x3^2+x2^2*x3+x2*x3^2)/(x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2*x3+x3^2))^(9/2)*x1^9+ 68068000*(x1*x2*x3*(x1^2*x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2*x3+x2*x3^2) /(x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2*x3+x3^2))^(9/2)*x2^9+68068000*(x1*x2*x3*(x1^2* x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2*x3+x2*x3^2)/(x1^2+2*x1*x2+2*x1*x3+x2 ^2+2*x2*x3+x3^2))^(9/2)*x3^9+4900896000*(x1*x2*x3*(x1^2*x2+x1^2*x3+x1*x2^2-3*x1 *x2*x3+x1*x3^2+x2^2*x3+x2*x3^2)/(x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2*x3+x3^2))^(9/2) *x1^7*x2*x3+17153136000*(x1*x2*x3*(x1^2*x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+x1*x3^2+ x2^2*x3+x2*x3^2)/(x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2*x3+x3^2))^(9/2)*x1^6*x2^2*x3+ 17153136000*(x1*x2*x3*(x1^2*x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2*x3+x2*x3 ^2)/(x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2*x3+x3^2))^(9/2)*x1^6*x2*x3^2+34306272000*( x1*x2*x3*(x1^2*x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2*x3+x2*x3^2)/(x1^2+2* x1*x2+2*x1*x3+x2^2+2*x2*x3+x3^2))^(9/2)*x1^5*x2^3*x3+51459408000*(x1*x2*x3*(x1^ 2*x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2*x3+x2*x3^2)/(x1^2+2*x1*x2+2*x1*x3+ x2^2+2*x2*x3+x3^2))^(9/2)*x1^5*x2^2*x3^2+34306272000*(x1*x2*x3*(x1^2*x2+x1^2*x3 +x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2*x3+x2*x3^2)/(x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2*x3 +x3^2))^(9/2)*x1^5*x2*x3^3+42882840000*(x1*x2*x3*(x1^2*x2+x1^2*x3+x1*x2^2-3*x1* x2*x3+x1*x3^2+x2^2*x3+x2*x3^2)/(x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2*x3+x3^2))^(9/2)* x1^4*x2^4*x3+85765680000*(x1*x2*x3*(x1^2*x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+x1*x3^2+ x2^2*x3+x2*x3^2)/(x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2*x3+x3^2))^(9/2)*x1^4*x2^3*x3^2 +85765680000*(x1*x2*x3*(x1^2*x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2*x3+x2* x3^2)/(x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2*x3+x3^2))^(9/2)*x1^4*x2^2*x3^3+ 42882840000*(x1*x2*x3*(x1^2*x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2*x3+x2*x3 ^2)/(x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2*x3+x3^2))^(9/2)*x1^4*x2*x3^4+34306272000*( x1*x2*x3*(x1^2*x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2*x3+x2*x3^2)/(x1^2+2* x1*x2+2*x1*x3+x2^2+2*x2*x3+x3^2))^(9/2)*x1^3*x2^5*x3+85765680000*(x1*x2*x3*(x1^ 2*x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2*x3+x2*x3^2)/(x1^2+2*x1*x2+2*x1*x3+ x2^2+2*x2*x3+x3^2))^(9/2)*x1^3*x2^4*x3^2+114354240000*(x1*x2*x3*(x1^2*x2+x1^2* x3+x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2*x3+x2*x3^2)/(x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2* x3+x3^2))^(9/2)*x1^3*x2^3*x3^3+85765680000*(x1*x2*x3*(x1^2*x2+x1^2*x3+x1*x2^2-3 *x1*x2*x3+x1*x3^2+x2^2*x3+x2*x3^2)/(x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2*x3+x3^2))^(9 /2)*x1^3*x2^2*x3^4+34306272000*(x1*x2*x3*(x1^2*x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+x1 *x3^2+x2^2*x3+x2*x3^2)/(x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2*x3+x3^2))^(9/2)*x1^3*x2* x3^5+17153136000*(x1*x2*x3*(x1^2*x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2*x3+ x2*x3^2)/(x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2*x3+x3^2))^(9/2)*x1^2*x2^6*x3+ 51459408000*(x1*x2*x3*(x1^2*x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2*x3+x2*x3 ^2)/(x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2*x3+x3^2))^(9/2)*x1^2*x2^5*x3^2+85765680000* (x1*x2*x3*(x1^2*x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2*x3+x2*x3^2)/(x1^2+2* x1*x2+2*x1*x3+x2^2+2*x2*x3+x3^2))^(9/2)*x1^2*x2^4*x3^3+85765680000*(x1*x2*x3*( x1^2*x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2*x3+x2*x3^2)/(x1^2+2*x1*x2+2*x1* x3+x2^2+2*x2*x3+x3^2))^(9/2)*x1^2*x2^3*x3^4+51459408000*(x1*x2*x3*(x1^2*x2+x1^2 *x3+x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2*x3+x2*x3^2)/(x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2 *x3+x3^2))^(9/2)*x1^2*x2^2*x3^5+17153136000*(x1*x2*x3*(x1^2*x2+x1^2*x3+x1*x2^2-\ 3*x1*x2*x3+x1*x3^2+x2^2*x3+x2*x3^2)/(x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2*x3+x3^2))^( 9/2)*x1^2*x2*x3^6+4900896000*(x1*x2*x3*(x1^2*x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+x1* x3^2+x2^2*x3+x2*x3^2)/(x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2*x3+x3^2))^(9/2)*x1*x2^7* x3+17153136000*(x1*x2*x3*(x1^2*x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2*x3+x2 *x3^2)/(x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2*x3+x3^2))^(9/2)*x1*x2^6*x3^2+34306272000 *(x1*x2*x3*(x1^2*x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2*x3+x2*x3^2)/(x1^2+2 *x1*x2+2*x1*x3+x2^2+2*x2*x3+x3^2))^(9/2)*x1*x2^5*x3^3+42882840000*(x1*x2*x3*(x1 ^2*x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2*x3+x2*x3^2)/(x1^2+2*x1*x2+2*x1*x3 +x2^2+2*x2*x3+x3^2))^(9/2)*x1*x2^4*x3^4+34306272000*(x1*x2*x3*(x1^2*x2+x1^2*x3+ x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2*x3+x2*x3^2)/(x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2*x3+ x3^2))^(9/2)*x1*x2^3*x3^5+17153136000*(x1*x2*x3*(x1^2*x2+x1^2*x3+x1*x2^2-3*x1* x2*x3+x1*x3^2+x2^2*x3+x2*x3^2)/(x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2*x3+x3^2))^(9/2)* x1*x2^2*x3^6+4900896000*(x1*x2*x3*(x1^2*x2+x1^2*x3+x1*x2^2-3*x1*x2*x3+x1*x3^2+ x2^2*x3+x2*x3^2)/(x1^2+2*x1*x2+2*x1*x3+x2^2+2*x2*x3+x3^2))^(9/2)*x1*x2*x3^7) The , 10, -th (straight) moment of the Duration is: 17 17 16 2 3 A B C (35532270 A B + 35532270 A C + 639580860 A B 16 16 2 15 3 + 1279161720 A B C + 639580860 A C + 5319539487 A B 15 2 15 2 15 3 + 16075516284 A B C + 16075516284 A B C + 5319539487 A C 14 4 14 3 14 2 2 + 27123967152 A B + 110366233776 A B C + 166484533248 A B C 14 3 14 4 13 5 + 110366233776 A B C + 27123967152 A C + 94854845880 A B 13 4 13 3 2 13 2 3 + 487994291280 A B C + 989862482880 A B C + 989862482880 A B C 13 4 13 5 12 6 + 487994291280 A B C + 94854845880 A C + 241131446640 A B 12 5 12 4 2 + 1507943984400 A B C + 3863746656000 A B C 12 3 3 12 2 4 + 5193868236480 A B C + 3863746656000 A B C 12 5 12 6 11 7 + 1507943984400 A B C + 241131446640 A C + 460757848662 A B 11 6 11 5 2 + 3410385797808 A B C + 10614956833152 A B C 11 4 3 11 3 4 + 18018354922848 A B C + 18018354922848 A B C 11 2 5 11 6 11 7 + 10614956833152 A B C + 3410385797808 A B C + 460757848662 A C 10 8 10 7 10 6 2 + 674864800200 A B + 5801438692872 A B C + 21362994473472 A B C 10 5 3 10 4 4 + 44038168122816 A B C + 55603495084032 A B C 10 3 5 10 2 6 + 44038168122816 A B C + 21362994473472 A B C 10 7 10 8 9 9 + 5801438692872 A B C + 674864800200 A C + 765584060508 A B 9 8 9 7 2 + 7539676703640 A B C + 32237429091264 A B C 9 6 3 9 5 4 + 78606114475968 A B C + 120518725969152 A B C 9 4 5 9 3 6 + 120518725969152 A B C + 78606114475968 A B C 9 2 7 9 8 9 9 + 32237429091264 A B C + 7539676703640 A B C + 765584060508 A C 8 10 8 9 8 8 2 + 674864800200 A B + 7539676703640 A B C + 36928917360000 A B C 8 7 3 8 6 4 + 104544693059760 A B C + 189582199418880 A B C 8 5 5 8 4 6 + 230203223631360 A B C + 189582199418880 A B C 8 3 7 8 2 8 + 104544693059760 A B C + 36928917360000 A B C 8 9 8 10 7 11 + 7539676703640 A B C + 674864800200 A C + 460757848662 A B 7 10 7 9 2 + 5801438692872 A B C + 32237429091264 A B C 7 8 3 7 7 4 + 104544693059760 A B C + 220102979885568 A B C 7 6 5 7 5 6 + 316102880381184 A B C + 316102880381184 A B C 7 4 7 7 3 8 + 220102979885568 A B C + 104544693059760 A B C 7 2 9 7 10 7 11 + 32237429091264 A B C + 5801438692872 A B C + 460757848662 A C 6 12 6 11 6 10 2 + 241131446640 A B + 3410385797808 A B C + 21362994473472 A B C 6 9 3 6 8 4 + 78606114475968 A B C + 189582199418880 A B C 6 7 5 6 6 6 + 316102880381184 A B C + 373866110631936 A B C 6 5 7 6 4 8 + 316102880381184 A B C + 189582199418880 A B C 6 3 9 6 2 10 + 78606114475968 A B C + 21362994473472 A B C 6 11 6 12 5 13 + 3410385797808 A B C + 241131446640 A C + 94854845880 A B 5 12 5 11 2 + 1507943984400 A B C + 10614956833152 A B C 5 10 3 5 9 4 + 44038168122816 A B C + 120518725969152 A B C 5 8 5 5 7 6 + 230203223631360 A B C + 316102880381184 A B C 5 6 7 5 5 8 + 316102880381184 A B C + 230203223631360 A B C 5 4 9 5 3 10 + 120518725969152 A B C + 44038168122816 A B C 5 2 11 5 12 5 13 + 10614956833152 A B C + 1507943984400 A B C + 94854845880 A C 4 14 4 13 4 12 2 + 27123967152 A B + 487994291280 A B C + 3863746656000 A B C 4 11 3 4 10 4 + 18018354922848 A B C + 55603495084032 A B C 4 9 5 4 8 6 + 120518725969152 A B C + 189582199418880 A B C 4 7 7 4 6 8 + 220102979885568 A B C + 189582199418880 A B C 4 5 9 4 4 10 + 120518725969152 A B C + 55603495084032 A B C 4 3 11 4 2 12 + 18018354922848 A B C + 3863746656000 A B C 4 13 4 14 3 15 + 487994291280 A B C + 27123967152 A C + 5319539487 A B 3 14 3 13 2 3 12 3 + 110366233776 A B C + 989862482880 A B C + 5193868236480 A B C 3 11 4 3 10 5 + 18018354922848 A B C + 44038168122816 A B C 3 9 6 3 8 7 + 78606114475968 A B C + 104544693059760 A B C 3 7 8 3 6 9 + 104544693059760 A B C + 78606114475968 A B C 3 5 10 3 4 11 + 44038168122816 A B C + 18018354922848 A B C 3 3 12 3 2 13 3 14 + 5193868236480 A B C + 989862482880 A B C + 110366233776 A B C 3 15 2 16 2 15 + 5319539487 A C + 639580860 A B + 16075516284 A B C 2 14 2 2 13 3 + 166484533248 A B C + 989862482880 A B C 2 12 4 2 11 5 + 3863746656000 A B C + 10614956833152 A B C 2 10 6 2 9 7 + 21362994473472 A B C + 32237429091264 A B C 2 8 8 2 7 9 + 36928917360000 A B C + 32237429091264 A B C 2 6 10 2 5 11 + 21362994473472 A B C + 10614956833152 A B C 2 4 12 2 3 13 + 3863746656000 A B C + 989862482880 A B C 2 2 14 2 15 2 16 + 166484533248 A B C + 16075516284 A B C + 639580860 A C 17 16 15 2 + 35532270 A B + 1279161720 A B C + 16075516284 A B C 14 3 13 4 12 5 + 110366233776 A B C + 487994291280 A B C + 1507943984400 A B C 11 6 10 7 9 8 + 3410385797808 A B C + 5801438692872 A B C + 7539676703640 A B C 8 9 7 10 6 11 + 7539676703640 A B C + 5801438692872 A B C + 3410385797808 A B C 5 12 4 13 3 14 + 1507943984400 A B C + 487994291280 A B C + 110366233776 A B C 2 15 16 17 + 16075516284 A B C + 1279161720 A B C + 35532270 A C 17 16 2 15 3 + 35532270 B C + 639580860 B C + 5319539487 B C 14 4 13 5 12 6 + 27123967152 B C + 94854845880 B C + 241131446640 B C 11 7 10 8 9 9 + 460757848662 B C + 674864800200 B C + 765584060508 B C 8 10 7 11 6 12 + 674864800200 B C + 460757848662 B C + 241131446640 B C 5 13 4 14 3 15 + 94854845880 B C + 27123967152 B C + 5319539487 B C 2 16 17 15 15 + 639580860 B C + 35532270 B C - 1052080407 A B - 1052080407 A C 14 2 14 14 2 - 16833286512 A B - 33666573024 A B C - 16833286512 A C 13 3 13 2 13 2 - 122788283940 A B - 371826216720 A B C - 371826216720 A B C 13 3 12 4 12 3 - 122788283940 A C - 540705919320 A B - 2211282785880 A B C 12 2 2 12 3 12 4 - 3341153733120 A B C - 2211282785880 A B C - 540705919320 A C 11 5 11 4 11 3 2 - 1604357950356 A B - 8327662326648 A B C - 16965753043680 A B C 11 2 3 11 4 11 5 - 16965753043680 A B C - 8327662326648 A B C - 1604357950356 A C 10 6 10 5 - 3390220180368 A B - 21491918333424 A B C 10 4 2 10 3 3 - 55510361496576 A B C - 74817326687040 A B C 10 2 4 10 5 - 55510361496576 A B C - 21491918333424 A B C 10 6 9 7 9 6 - 3390220180368 A C - 5257703333862 A B - 39677024327472 A B C 9 5 2 9 4 3 - 125067245038848 A B C - 213623750940768 A B C 9 3 4 9 2 5 - 213623750940768 A B C - 125067245038848 A B C 9 6 9 7 8 8 - 39677024327472 A B C - 5257703333862 A C - 6076285494300 A B 8 7 8 6 2 - 53631801393660 A B C - 201160240404480 A B C 8 5 3 8 4 4 - 419207458229280 A B C - 531205467448320 A B C 8 3 5 8 2 6 - 419207458229280 A B C - 201160240404480 A B C 8 7 8 8 7 9 - 53631801393660 A B C - 6076285494300 A C - 5257703333862 A B 7 8 7 7 2 - 53631801393660 A B C - 235247529242880 A B C 7 6 3 7 5 4 - 583261091249760 A B C - 901650321500544 A B C 7 4 5 7 3 6 - 901650321500544 A B C - 583261091249760 A B C 7 2 7 7 8 7 9 - 235247529242880 A B C - 53631801393660 A B C - 5257703333862 A C 6 10 6 9 6 8 2 - 3390220180368 A B - 39677024327472 A B C - 201160240404480 A B C 6 7 3 6 6 4 - 583261091249760 A B C - 1072690399807488 A B C 6 5 5 6 4 6 - 1308605489630208 A B C - 1072690399807488 A B C 6 3 7 6 2 8 - 583261091249760 A B C - 201160240404480 A B C 6 9 6 10 5 11 - 39677024327472 A B C - 3390220180368 A C - 1604357950356 A B 5 10 5 9 2 - 21491918333424 A B C - 125067245038848 A B C 5 8 3 5 7 4 - 419207458229280 A B C - 901650321500544 A B C 5 6 5 5 5 6 - 1308605489630208 A B C - 1308605489630208 A B C 5 4 7 5 3 8 - 901650321500544 A B C - 419207458229280 A B C 5 2 9 5 10 - 125067245038848 A B C - 21491918333424 A B C 5 11 4 12 4 11 - 1604357950356 A C - 540705919320 A B - 8327662326648 A B C 4 10 2 4 9 3 - 55510361496576 A B C - 213623750940768 A B C 4 8 4 4 7 5 - 531205467448320 A B C - 901650321500544 A B C 4 6 6 4 5 7 - 1072690399807488 A B C - 901650321500544 A B C 4 4 8 4 3 9 - 531205467448320 A B C - 213623750940768 A B C 4 2 10 4 11 4 12 - 55510361496576 A B C - 8327662326648 A B C - 540705919320 A C 3 13 3 12 3 11 2 - 122788283940 A B - 2211282785880 A B C - 16965753043680 A B C 3 10 3 3 9 4 - 74817326687040 A B C - 213623750940768 A B C 3 8 5 3 7 6 - 419207458229280 A B C - 583261091249760 A B C 3 6 7 3 5 8 - 583261091249760 A B C - 419207458229280 A B C 3 4 9 3 3 10 - 213623750940768 A B C - 74817326687040 A B C 3 2 11 3 12 3 13 - 16965753043680 A B C - 2211282785880 A B C - 122788283940 A C 2 14 2 13 2 12 2 - 16833286512 A B - 371826216720 A B C - 3341153733120 A B C 2 11 3 2 10 4 - 16965753043680 A B C - 55510361496576 A B C 2 9 5 2 8 6 - 125067245038848 A B C - 201160240404480 A B C 2 7 7 2 6 8 - 235247529242880 A B C - 201160240404480 A B C 2 5 9 2 4 10 - 125067245038848 A B C - 55510361496576 A B C 2 3 11 2 2 12 - 16965753043680 A B C - 3341153733120 A B C 2 13 2 14 15 - 371826216720 A B C - 16833286512 A C - 1052080407 A B 14 13 2 12 3 - 33666573024 A B C - 371826216720 A B C - 2211282785880 A B C 11 4 10 5 - 8327662326648 A B C - 21491918333424 A B C 9 6 8 7 7 8 - 39677024327472 A B C - 53631801393660 A B C - 53631801393660 A B C 6 9 5 10 - 39677024327472 A B C - 21491918333424 A B C 4 11 3 12 2 13 - 8327662326648 A B C - 2211282785880 A B C - 371826216720 A B C 14 15 15 - 33666573024 A B C - 1052080407 A C - 1052080407 B C 14 2 13 3 12 4 - 16833286512 B C - 122788283940 B C - 540705919320 B C 11 5 10 6 9 7 - 1604357950356 B C - 3390220180368 B C - 5257703333862 B C 8 8 7 9 6 10 - 6076285494300 B C - 5257703333862 B C - 3390220180368 B C 5 11 4 12 3 13 - 1604357950356 B C - 540705919320 B C - 122788283940 B C 2 14 15 13 - 16833286512 B C - 1052080407 B C + 11589162480 A B 13 12 2 12 + 11589162480 A C + 162248274720 A B + 324496549440 A B C 12 2 11 3 11 2 + 162248274720 A C + 1016479922472 A B + 3087573630624 A B C 11 2 11 3 10 4 + 3087573630624 A B C + 1016479922472 A C + 3760848784224 A B 10 3 10 2 2 + 15501001495392 A B C + 23480305422336 A B C 10 3 10 4 9 5 + 15501001495392 A B C + 3760848784224 A C + 9134136374832 A B 9 4 9 3 2 + 48087077437728 A B C + 98640770143104 A B C 9 2 3 9 4 9 5 + 98640770143104 A B C + 48087077437728 A B C + 9134136374832 A C 8 6 8 5 8 4 2 + 15314250420000 A B + 99270558344160 A B C + 259756176637440 A B C 8 3 3 8 2 4 + 351599737426560 A B C + 259756176637440 A B C 8 5 8 6 7 7 + 99270558344160 A B C + 15314250420000 A C + 18150366818052 A B 7 6 7 5 2 + 141515594162208 A B C + 455614347025152 A B C 7 4 3 7 3 4 + 786353150495808 A B C + 786353150495808 A B C 7 2 5 7 6 + 455614347025152 A B C + 141515594162208 A B C 7 7 6 8 6 7 + 18150366818052 A C + 15314250420000 A B + 141515594162208 A B C 6 6 2 6 5 3 + 547887920928768 A B C + 1163223443015424 A B C 6 4 4 6 3 5 + 1483073731657728 A B C + 1163223443015424 A B C 6 2 6 6 7 + 547887920928768 A B C + 141515594162208 A B C 6 8 5 9 5 8 + 15314250420000 A C + 9134136374832 A B + 99270558344160 A B C 5 7 2 5 6 3 + 455614347025152 A B C + 1163223443015424 A B C 5 5 4 5 4 5 + 1824301058291712 A B C + 1824301058291712 A B C 5 3 6 5 2 7 + 1163223443015424 A B C + 455614347025152 A B C 5 8 5 9 4 10 + 99270558344160 A B C + 9134136374832 A C + 3760848784224 A B 4 9 4 8 2 + 48087077437728 A B C + 259756176637440 A B C 4 7 3 4 6 4 + 786353150495808 A B C + 1483073731657728 A B C 4 5 5 4 4 6 + 1824301058291712 A B C + 1483073731657728 A B C 4 3 7 4 2 8 + 786353150495808 A B C + 259756176637440 A B C 4 9 4 10 3 11 + 48087077437728 A B C + 3760848784224 A C + 1016479922472 A B 3 10 3 9 2 + 15501001495392 A B C + 98640770143104 A B C 3 8 3 3 7 4 + 351599737426560 A B C + 786353150495808 A B C 3 6 5 3 5 6 + 1163223443015424 A B C + 1163223443015424 A B C 3 4 7 3 3 8 + 786353150495808 A B C + 351599737426560 A B C 3 2 9 3 10 3 11 + 98640770143104 A B C + 15501001495392 A B C + 1016479922472 A C 2 12 2 11 2 10 2 + 162248274720 A B + 3087573630624 A B C + 23480305422336 A B C 2 9 3 2 8 4 + 98640770143104 A B C + 259756176637440 A B C 2 7 5 2 6 6 + 455614347025152 A B C + 547887920928768 A B C 2 5 7 2 4 8 + 455614347025152 A B C + 259756176637440 A B C 2 3 9 2 2 10 + 98640770143104 A B C + 23480305422336 A B C 2 11 2 12 13 + 3087573630624 A B C + 162248274720 A C + 11589162480 A B 12 11 2 10 3 + 324496549440 A B C + 3087573630624 A B C + 15501001495392 A B C 9 4 8 5 + 48087077437728 A B C + 99270558344160 A B C 7 6 6 7 + 141515594162208 A B C + 141515594162208 A B C 5 8 4 9 + 99270558344160 A B C + 48087077437728 A B C 3 10 2 11 12 + 15501001495392 A B C + 3087573630624 A B C + 324496549440 A B C 13 13 12 2 + 11589162480 A C + 11589162480 B C + 162248274720 B C 11 3 10 4 9 5 + 1016479922472 B C + 3760848784224 B C + 9134136374832 B C 8 6 7 7 6 8 + 15314250420000 B C + 18150366818052 B C + 15314250420000 B C 5 9 4 10 3 11 + 9134136374832 B C + 3760848784224 B C + 1016479922472 B C 2 12 13 11 + 162248274720 B C + 11589162480 B C - 57116427858 A B 11 10 2 10 - 57116427858 A C - 685397134296 A B - 1370794268592 A B C 10 2 9 3 9 2 - 685397134296 A C - 3581638112178 A B - 10932960462984 A B C 9 2 9 3 8 4 - 10932960462984 A B C - 3581638112178 A C - 10685152864260 A B 8 3 8 2 2 - 44621072721540 A B C - 67871839714560 A B C 8 3 8 4 7 5 - 44621072721540 A B C - 10685152864260 A C - 20058776986374 A B 7 4 7 3 2 - 108259518848292 A B C - 224732892499920 A B C 7 2 3 7 4 - 224732892499920 A B C - 108259518848292 A B C 7 5 6 6 6 5 - 20058776986374 A C - 24656442784848 A B - 166532657692464 A B C 6 4 2 6 3 3 - 446208412801536 A B C - 608664395787840 A B C 6 2 4 6 5 - 446208412801536 A B C - 166532657692464 A B C 6 6 5 7 5 6 - 24656442784848 A C - 20058776986374 A B - 166532657692464 A B C 5 5 2 5 4 3 - 558199593374976 A B C - 982524231200736 A B C 5 3 4 5 2 5 - 982524231200736 A B C - 558199593374976 A B C 5 6 5 7 4 8 - 166532657692464 A B C - 20058776986374 A C - 10685152864260 A B 4 7 4 6 2 - 108259518848292 A B C - 446208412801536 A B C 4 5 3 4 4 4 - 982524231200736 A B C - 1267780368766464 A B C 4 3 5 4 2 6 - 982524231200736 A B C - 446208412801536 A B C 4 7 4 8 3 9 - 108259518848292 A B C - 10685152864260 A C - 3581638112178 A B 3 8 3 7 2 - 44621072721540 A B C - 224732892499920 A B C 3 6 3 3 5 4 - 608664395787840 A B C - 982524231200736 A B C 3 4 5 3 3 6 - 982524231200736 A B C - 608664395787840 A B C 3 2 7 3 8 3 9 - 224732892499920 A B C - 44621072721540 A B C - 3581638112178 A C 2 10 2 9 2 8 2 - 685397134296 A B - 10932960462984 A B C - 67871839714560 A B C 2 7 3 2 6 4 - 224732892499920 A B C - 446208412801536 A B C 2 5 5 2 4 6 - 558199593374976 A B C - 446208412801536 A B C 2 3 7 2 2 8 - 224732892499920 A B C - 67871839714560 A B C 2 9 2 10 11 - 10932960462984 A B C - 685397134296 A C - 57116427858 A B 10 9 2 8 3 - 1370794268592 A B C - 10932960462984 A B C - 44621072721540 A B C 7 4 6 5 - 108259518848292 A B C - 166532657692464 A B C 5 6 4 7 - 166532657692464 A B C - 108259518848292 A B C 3 8 2 9 10 - 44621072721540 A B C - 10932960462984 A B C - 1370794268592 A B C 11 11 10 2 - 57116427858 A C - 57116427858 B C - 685397134296 B C 9 3 8 4 7 5 - 3581638112178 B C - 10685152864260 B C - 20058776986374 B C 6 6 5 7 4 8 - 24656442784848 B C - 20058776986374 B C - 10685152864260 B C 3 9 2 10 11 - 3581638112178 B C - 685397134296 B C - 57116427858 B C 9 9 8 2 + 115319614410 A B + 115319614410 A C + 1153196144100 A B 8 8 2 7 3 + 2306392288200 A B C + 1153196144100 A C + 4808827920897 A B 7 2 7 2 7 3 + 14807038490244 A B C + 14807038490244 A B C + 4808827920897 A C 6 4 6 3 6 2 2 + 10793915908776 A B + 46220101455528 A B C + 70852371093504 A B C 6 3 6 4 5 5 + 46220101455528 A B C + 10793915908776 A C + 14068992958020 A B 5 4 5 3 2 + 79985684554776 A B C + 170119495177632 A B C 5 2 3 5 4 5 5 + 170119495177632 A B C + 79985684554776 A B C + 14068992958020 A C 4 6 4 5 4 4 2 + 10793915908776 A B + 79985684554776 A B C + 225841932860544 A B C 4 3 3 4 2 4 + 313300328429088 A B C + 225841932860544 A B C 4 5 4 6 3 7 + 79985684554776 A B C + 10793915908776 A C + 4808827920897 A B 3 6 3 5 2 + 46220101455528 A B C + 170119495177632 A B C 3 4 3 3 3 4 + 313300328429088 A B C + 313300328429088 A B C 3 2 5 3 6 3 7 + 170119495177632 A B C + 46220101455528 A B C + 4808827920897 A C 2 8 2 7 2 6 2 + 1153196144100 A B + 14807038490244 A B C + 70852371093504 A B C 2 5 3 2 4 4 + 170119495177632 A B C + 225841932860544 A B C 2 3 5 2 2 6 + 170119495177632 A B C + 70852371093504 A B C 2 7 2 8 9 + 14807038490244 A B C + 1153196144100 A C + 115319614410 A B 8 7 2 6 3 + 2306392288200 A B C + 14807038490244 A B C + 46220101455528 A B C 5 4 4 5 3 6 + 79985684554776 A B C + 79985684554776 A B C + 46220101455528 A B C 2 7 8 9 + 14807038490244 A B C + 2306392288200 A B C + 115319614410 A C 9 8 2 7 3 + 115319614410 B C + 1153196144100 B C + 4808827920897 B C 6 4 5 5 4 6 + 10793915908776 B C + 14068992958020 B C + 10793915908776 B C 3 7 2 8 9 + 4808827920897 B C + 1153196144100 B C + 115319614410 B C 7 7 6 2 - 55842438015 A B - 55842438015 A C - 446739504120 A B 6 6 2 5 3 - 893479008240 A B C - 446739504120 A C - 1377446804370 A B 5 2 5 2 5 3 - 4318481873160 A B C - 4318481873160 A B C - 1377446804370 A C 4 4 4 3 4 2 2 - 2010327768540 A B - 9158159834460 A B C - 14295664131840 A B C 4 3 4 4 3 5 - 9158159834460 A B C - 2010327768540 A C - 1377446804370 A B 3 4 3 3 2 - 9158159834460 A B C - 20847843525600 A B C 3 2 3 3 4 3 5 - 20847843525600 A B C - 9158159834460 A B C - 1377446804370 A C 2 6 2 5 2 4 2 - 446739504120 A B - 4318481873160 A B C - 14295664131840 A B C 2 3 3 2 2 4 - 20847843525600 A B C - 14295664131840 A B C 2 5 2 6 7 - 4318481873160 A B C - 446739504120 A C - 55842438015 A B 6 5 2 4 3 - 893479008240 A B C - 4318481873160 A B C - 9158159834460 A B C 3 4 2 5 6 - 9158159834460 A B C - 4318481873160 A B C - 893479008240 A B C 7 7 6 2 - 55842438015 A C - 55842438015 B C - 446739504120 B C 5 3 4 4 3 5 - 1377446804370 B C - 2010327768540 B C - 1377446804370 B C 2 6 7 5 - 446739504120 B C - 55842438015 B C + 1844541888 A B 5 4 2 4 + 1844541888 A C + 11067251328 A B + 22134502656 A B C 4 2 3 3 3 2 + 11067251328 A C + 21212231712 A B + 70092591744 A B C 3 2 3 3 2 4 + 70092591744 A B C + 21212231712 A C + 11067251328 A B 2 3 2 2 2 2 3 + 70092591744 A B C + 118050680832 A B C + 70092591744 A B C 2 4 5 4 + 11067251328 A C + 1844541888 A B + 22134502656 A B C 3 2 2 3 4 + 70092591744 A B C + 70092591744 A B C + 22134502656 A B C 5 5 4 2 + 1844541888 A C + 1844541888 B C + 11067251328 B C 3 3 2 4 5 + 21212231712 B C + 11067251328 B C + 1844541888 B C 3 3 2 2 + 23845772160 A B + 23845772160 A C + 95383088640 A B 2 2 2 3 + 190766177280 A B C + 95383088640 A C + 23845772160 A B 2 2 3 + 190766177280 A B C + 190766177280 A B C + 23845772160 A C 3 2 2 3 + 23845772160 B C + 95383088640 B C + 23845772160 B C + 96321757952 A B + 96321757952 A C + 96321757952 B C + 56313241600)/( 57939481600 (A + B + C)) and in Maple notation 3/57939481600*A*B*C*(35532270*A^17*B+35532270*A^17*C+639580860*A^16*B^2+ 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*B^7*C^7+189582199418880*A^4*B^6*C^8+120518725969152*A^4*B^5*C^9+55603495084032 *A^4*B^4*C^10+18018354922848*A^4*B^3*C^11+3863746656000*A^4*B^2*C^12+ 487994291280*A^4*B*C^13+27123967152*A^4*C^14+5319539487*A^3*B^15+110366233776*A ^3*B^14*C+989862482880*A^3*B^13*C^2+5193868236480*A^3*B^12*C^3+18018354922848*A ^3*B^11*C^4+44038168122816*A^3*B^10*C^5+78606114475968*A^3*B^9*C^6+ 104544693059760*A^3*B^8*C^7+104544693059760*A^3*B^7*C^8+78606114475968*A^3*B^6* C^9+44038168122816*A^3*B^5*C^10+18018354922848*A^3*B^4*C^11+5193868236480*A^3*B ^3*C^12+989862482880*A^3*B^2*C^13+110366233776*A^3*B*C^14+5319539487*A^3*C^15+ 639580860*A^2*B^16+16075516284*A^2*B^15*C+166484533248*A^2*B^14*C^2+ 989862482880*A^2*B^13*C^3+3863746656000*A^2*B^12*C^4+10614956833152*A^2*B^11*C^ 5+21362994473472*A^2*B^10*C^6+32237429091264*A^2*B^9*C^7+36928917360000*A^2*B^8 *C^8+32237429091264*A^2*B^7*C^9+21362994473472*A^2*B^6*C^10+10614956833152*A^2* B^5*C^11+3863746656000*A^2*B^4*C^12+989862482880*A^2*B^3*C^13+166484533248*A^2* B^2*C^14+16075516284*A^2*B*C^15+639580860*A^2*C^16+35532270*A*B^17+1279161720*A *B^16*C+16075516284*A*B^15*C^2+110366233776*A*B^14*C^3+487994291280*A*B^13*C^4+ 1507943984400*A*B^12*C^5+3410385797808*A*B^11*C^6+5801438692872*A*B^10*C^7+ 7539676703640*A*B^9*C^8+7539676703640*A*B^8*C^9+5801438692872*A*B^7*C^10+ 3410385797808*A*B^6*C^11+1507943984400*A*B^5*C^12+487994291280*A*B^4*C^13+ 110366233776*A*B^3*C^14+16075516284*A*B^2*C^15+1279161720*A*B*C^16+35532270*A*C ^17+35532270*B^17*C+639580860*B^16*C^2+5319539487*B^15*C^3+27123967152*B^14*C^4 +94854845880*B^13*C^5+241131446640*B^12*C^6+460757848662*B^11*C^7+674864800200* B^10*C^8+765584060508*B^9*C^9+674864800200*B^8*C^10+460757848662*B^7*C^11+ 241131446640*B^6*C^12+94854845880*B^5*C^13+27123967152*B^4*C^14+5319539487*B^3* C^15+639580860*B^2*C^16+35532270*B*C^17-1052080407*A^15*B-1052080407*A^15*C-\ 16833286512*A^14*B^2-33666573024*A^14*B*C-16833286512*A^14*C^2-122788283940*A^ 13*B^3-371826216720*A^13*B^2*C-371826216720*A^13*B*C^2-122788283940*A^13*C^3-\ 540705919320*A^12*B^4-2211282785880*A^12*B^3*C-3341153733120*A^12*B^2*C^2-\ 2211282785880*A^12*B*C^3-540705919320*A^12*C^4-1604357950356*A^11*B^5-\ 8327662326648*A^11*B^4*C-16965753043680*A^11*B^3*C^2-16965753043680*A^11*B^2*C^ 3-8327662326648*A^11*B*C^4-1604357950356*A^11*C^5-3390220180368*A^10*B^6-\ 21491918333424*A^10*B^5*C-55510361496576*A^10*B^4*C^2-74817326687040*A^10*B^3*C ^3-55510361496576*A^10*B^2*C^4-21491918333424*A^10*B*C^5-3390220180368*A^10*C^6 -5257703333862*A^9*B^7-39677024327472*A^9*B^6*C-125067245038848*A^9*B^5*C^2-\ 213623750940768*A^9*B^4*C^3-213623750940768*A^9*B^3*C^4-125067245038848*A^9*B^2 *C^5-39677024327472*A^9*B*C^6-5257703333862*A^9*C^7-6076285494300*A^8*B^8-\ 53631801393660*A^8*B^7*C-201160240404480*A^8*B^6*C^2-419207458229280*A^8*B^5*C^ 3-531205467448320*A^8*B^4*C^4-419207458229280*A^8*B^3*C^5-201160240404480*A^8*B ^2*C^6-53631801393660*A^8*B*C^7-6076285494300*A^8*C^8-5257703333862*A^7*B^9-\ 53631801393660*A^7*B^8*C-235247529242880*A^7*B^7*C^2-583261091249760*A^7*B^6*C^ 3-901650321500544*A^7*B^5*C^4-901650321500544*A^7*B^4*C^5-583261091249760*A^7*B ^3*C^6-235247529242880*A^7*B^2*C^7-53631801393660*A^7*B*C^8-5257703333862*A^7*C ^9-3390220180368*A^6*B^10-39677024327472*A^6*B^9*C-201160240404480*A^6*B^8*C^2-\ 583261091249760*A^6*B^7*C^3-1072690399807488*A^6*B^6*C^4-1308605489630208*A^6*B ^5*C^5-1072690399807488*A^6*B^4*C^6-583261091249760*A^6*B^3*C^7-201160240404480 *A^6*B^2*C^8-39677024327472*A^6*B*C^9-3390220180368*A^6*C^10-1604357950356*A^5* B^11-21491918333424*A^5*B^10*C-125067245038848*A^5*B^9*C^2-419207458229280*A^5* B^8*C^3-901650321500544*A^5*B^7*C^4-1308605489630208*A^5*B^6*C^5-\ 1308605489630208*A^5*B^5*C^6-901650321500544*A^5*B^4*C^7-419207458229280*A^5*B^ 3*C^8-125067245038848*A^5*B^2*C^9-21491918333424*A^5*B*C^10-1604357950356*A^5*C ^11-540705919320*A^4*B^12-8327662326648*A^4*B^11*C-55510361496576*A^4*B^10*C^2-\ 213623750940768*A^4*B^9*C^3-531205467448320*A^4*B^8*C^4-901650321500544*A^4*B^7 *C^5-1072690399807488*A^4*B^6*C^6-901650321500544*A^4*B^5*C^7-531205467448320*A ^4*B^4*C^8-213623750940768*A^4*B^3*C^9-55510361496576*A^4*B^2*C^10-\ 8327662326648*A^4*B*C^11-540705919320*A^4*C^12-122788283940*A^3*B^13-\ 2211282785880*A^3*B^12*C-16965753043680*A^3*B^11*C^2-74817326687040*A^3*B^10*C^ 3-213623750940768*A^3*B^9*C^4-419207458229280*A^3*B^8*C^5-583261091249760*A^3*B ^7*C^6-583261091249760*A^3*B^6*C^7-419207458229280*A^3*B^5*C^8-213623750940768* A^3*B^4*C^9-74817326687040*A^3*B^3*C^10-16965753043680*A^3*B^2*C^11-\ 2211282785880*A^3*B*C^12-122788283940*A^3*C^13-16833286512*A^2*B^14-\ 371826216720*A^2*B^13*C-3341153733120*A^2*B^12*C^2-16965753043680*A^2*B^11*C^3-\ 55510361496576*A^2*B^10*C^4-125067245038848*A^2*B^9*C^5-201160240404480*A^2*B^8 *C^6-235247529242880*A^2*B^7*C^7-201160240404480*A^2*B^6*C^8-125067245038848*A^ 2*B^5*C^9-55510361496576*A^2*B^4*C^10-16965753043680*A^2*B^3*C^11-3341153733120 *A^2*B^2*C^12-371826216720*A^2*B*C^13-16833286512*A^2*C^14-1052080407*A*B^15-\ 33666573024*A*B^14*C-371826216720*A*B^13*C^2-2211282785880*A*B^12*C^3-\ 8327662326648*A*B^11*C^4-21491918333424*A*B^10*C^5-39677024327472*A*B^9*C^6-\ 53631801393660*A*B^8*C^7-53631801393660*A*B^7*C^8-39677024327472*A*B^6*C^9-\ 21491918333424*A*B^5*C^10-8327662326648*A*B^4*C^11-2211282785880*A*B^3*C^12-\ 371826216720*A*B^2*C^13-33666573024*A*B*C^14-1052080407*A*C^15-1052080407*B^15* C-16833286512*B^14*C^2-122788283940*B^13*C^3-540705919320*B^12*C^4-\ 1604357950356*B^11*C^5-3390220180368*B^10*C^6-5257703333862*B^9*C^7-\ 6076285494300*B^8*C^8-5257703333862*B^7*C^9-3390220180368*B^6*C^10-\ 1604357950356*B^5*C^11-540705919320*B^4*C^12-122788283940*B^3*C^13-16833286512* B^2*C^14-1052080407*B*C^15+11589162480*A^13*B+11589162480*A^13*C+162248274720*A ^12*B^2+324496549440*A^12*B*C+162248274720*A^12*C^2+1016479922472*A^11*B^3+ 3087573630624*A^11*B^2*C+3087573630624*A^11*B*C^2+1016479922472*A^11*C^3+ 3760848784224*A^10*B^4+15501001495392*A^10*B^3*C+23480305422336*A^10*B^2*C^2+ 15501001495392*A^10*B*C^3+3760848784224*A^10*C^4+9134136374832*A^9*B^5+ 48087077437728*A^9*B^4*C+98640770143104*A^9*B^3*C^2+98640770143104*A^9*B^2*C^3+ 48087077437728*A^9*B*C^4+9134136374832*A^9*C^5+15314250420000*A^8*B^6+ 99270558344160*A^8*B^5*C+259756176637440*A^8*B^4*C^2+351599737426560*A^8*B^3*C^ 3+259756176637440*A^8*B^2*C^4+99270558344160*A^8*B*C^5+15314250420000*A^8*C^6+ 18150366818052*A^7*B^7+141515594162208*A^7*B^6*C+455614347025152*A^7*B^5*C^2+ 786353150495808*A^7*B^4*C^3+786353150495808*A^7*B^3*C^4+455614347025152*A^7*B^2 *C^5+141515594162208*A^7*B*C^6+18150366818052*A^7*C^7+15314250420000*A^6*B^8+ 141515594162208*A^6*B^7*C+547887920928768*A^6*B^6*C^2+1163223443015424*A^6*B^5* C^3+1483073731657728*A^6*B^4*C^4+1163223443015424*A^6*B^3*C^5+547887920928768*A ^6*B^2*C^6+141515594162208*A^6*B*C^7+15314250420000*A^6*C^8+9134136374832*A^5*B ^9+99270558344160*A^5*B^8*C+455614347025152*A^5*B^7*C^2+1163223443015424*A^5*B^ 6*C^3+1824301058291712*A^5*B^5*C^4+1824301058291712*A^5*B^4*C^5+ 1163223443015424*A^5*B^3*C^6+455614347025152*A^5*B^2*C^7+99270558344160*A^5*B*C ^8+9134136374832*A^5*C^9+3760848784224*A^4*B^10+48087077437728*A^4*B^9*C+ 259756176637440*A^4*B^8*C^2+786353150495808*A^4*B^7*C^3+1483073731657728*A^4*B^ 6*C^4+1824301058291712*A^4*B^5*C^5+1483073731657728*A^4*B^4*C^6+786353150495808 *A^4*B^3*C^7+259756176637440*A^4*B^2*C^8+48087077437728*A^4*B*C^9+3760848784224 *A^4*C^10+1016479922472*A^3*B^11+15501001495392*A^3*B^10*C+98640770143104*A^3*B ^9*C^2+351599737426560*A^3*B^8*C^3+786353150495808*A^3*B^7*C^4+1163223443015424 *A^3*B^6*C^5+1163223443015424*A^3*B^5*C^6+786353150495808*A^3*B^4*C^7+ 351599737426560*A^3*B^3*C^8+98640770143104*A^3*B^2*C^9+15501001495392*A^3*B*C^ 10+1016479922472*A^3*C^11+162248274720*A^2*B^12+3087573630624*A^2*B^11*C+ 23480305422336*A^2*B^10*C^2+98640770143104*A^2*B^9*C^3+259756176637440*A^2*B^8* C^4+455614347025152*A^2*B^7*C^5+547887920928768*A^2*B^6*C^6+455614347025152*A^2 *B^5*C^7+259756176637440*A^2*B^4*C^8+98640770143104*A^2*B^3*C^9+23480305422336* A^2*B^2*C^10+3087573630624*A^2*B*C^11+162248274720*A^2*C^12+11589162480*A*B^13+ 324496549440*A*B^12*C+3087573630624*A*B^11*C^2+15501001495392*A*B^10*C^3+ 48087077437728*A*B^9*C^4+99270558344160*A*B^8*C^5+141515594162208*A*B^7*C^6+ 141515594162208*A*B^6*C^7+99270558344160*A*B^5*C^8+48087077437728*A*B^4*C^9+ 15501001495392*A*B^3*C^10+3087573630624*A*B^2*C^11+324496549440*A*B*C^12+ 11589162480*A*C^13+11589162480*B^13*C+162248274720*B^12*C^2+1016479922472*B^11* C^3+3760848784224*B^10*C^4+9134136374832*B^9*C^5+15314250420000*B^8*C^6+ 18150366818052*B^7*C^7+15314250420000*B^6*C^8+9134136374832*B^5*C^9+ 3760848784224*B^4*C^10+1016479922472*B^3*C^11+162248274720*B^2*C^12+11589162480 *B*C^13-57116427858*A^11*B-57116427858*A^11*C-685397134296*A^10*B^2-\ 1370794268592*A^10*B*C-685397134296*A^10*C^2-3581638112178*A^9*B^3-\ 10932960462984*A^9*B^2*C-10932960462984*A^9*B*C^2-3581638112178*A^9*C^3-\ 10685152864260*A^8*B^4-44621072721540*A^8*B^3*C-67871839714560*A^8*B^2*C^2-\ 44621072721540*A^8*B*C^3-10685152864260*A^8*C^4-20058776986374*A^7*B^5-\ 108259518848292*A^7*B^4*C-224732892499920*A^7*B^3*C^2-224732892499920*A^7*B^2*C ^3-108259518848292*A^7*B*C^4-20058776986374*A^7*C^5-24656442784848*A^6*B^6-\ 166532657692464*A^6*B^5*C-446208412801536*A^6*B^4*C^2-608664395787840*A^6*B^3*C ^3-446208412801536*A^6*B^2*C^4-166532657692464*A^6*B*C^5-24656442784848*A^6*C^6 -20058776986374*A^5*B^7-166532657692464*A^5*B^6*C-558199593374976*A^5*B^5*C^2-\ 982524231200736*A^5*B^4*C^3-982524231200736*A^5*B^3*C^4-558199593374976*A^5*B^2 *C^5-166532657692464*A^5*B*C^6-20058776986374*A^5*C^7-10685152864260*A^4*B^8-\ 108259518848292*A^4*B^7*C-446208412801536*A^4*B^6*C^2-982524231200736*A^4*B^5*C ^3-1267780368766464*A^4*B^4*C^4-982524231200736*A^4*B^3*C^5-446208412801536*A^4 *B^2*C^6-108259518848292*A^4*B*C^7-10685152864260*A^4*C^8-3581638112178*A^3*B^9 -44621072721540*A^3*B^8*C-224732892499920*A^3*B^7*C^2-608664395787840*A^3*B^6*C ^3-982524231200736*A^3*B^5*C^4-982524231200736*A^3*B^4*C^5-608664395787840*A^3* B^3*C^6-224732892499920*A^3*B^2*C^7-44621072721540*A^3*B*C^8-3581638112178*A^3* C^9-685397134296*A^2*B^10-10932960462984*A^2*B^9*C-67871839714560*A^2*B^8*C^2-\ 224732892499920*A^2*B^7*C^3-446208412801536*A^2*B^6*C^4-558199593374976*A^2*B^5 *C^5-446208412801536*A^2*B^4*C^6-224732892499920*A^2*B^3*C^7-67871839714560*A^2 *B^2*C^8-10932960462984*A^2*B*C^9-685397134296*A^2*C^10-57116427858*A*B^11-\ 1370794268592*A*B^10*C-10932960462984*A*B^9*C^2-44621072721540*A*B^8*C^3-\ 108259518848292*A*B^7*C^4-166532657692464*A*B^6*C^5-166532657692464*A*B^5*C^6-\ 108259518848292*A*B^4*C^7-44621072721540*A*B^3*C^8-10932960462984*A*B^2*C^9-\ 1370794268592*A*B*C^10-57116427858*A*C^11-57116427858*B^11*C-685397134296*B^10* C^2-3581638112178*B^9*C^3-10685152864260*B^8*C^4-20058776986374*B^7*C^5-\ 24656442784848*B^6*C^6-20058776986374*B^5*C^7-10685152864260*B^4*C^8-\ 3581638112178*B^3*C^9-685397134296*B^2*C^10-57116427858*B*C^11+115319614410*A^9 *B+115319614410*A^9*C+1153196144100*A^8*B^2+2306392288200*A^8*B*C+1153196144100 *A^8*C^2+4808827920897*A^7*B^3+14807038490244*A^7*B^2*C+14807038490244*A^7*B*C^ 2+4808827920897*A^7*C^3+10793915908776*A^6*B^4+46220101455528*A^6*B^3*C+ 70852371093504*A^6*B^2*C^2+46220101455528*A^6*B*C^3+10793915908776*A^6*C^4+ 14068992958020*A^5*B^5+79985684554776*A^5*B^4*C+170119495177632*A^5*B^3*C^2+ 170119495177632*A^5*B^2*C^3+79985684554776*A^5*B*C^4+14068992958020*A^5*C^5+ 10793915908776*A^4*B^6+79985684554776*A^4*B^5*C+225841932860544*A^4*B^4*C^2+ 313300328429088*A^4*B^3*C^3+225841932860544*A^4*B^2*C^4+79985684554776*A^4*B*C^ 5+10793915908776*A^4*C^6+4808827920897*A^3*B^7+46220101455528*A^3*B^6*C+ 170119495177632*A^3*B^5*C^2+313300328429088*A^3*B^4*C^3+313300328429088*A^3*B^3 *C^4+170119495177632*A^3*B^2*C^5+46220101455528*A^3*B*C^6+4808827920897*A^3*C^7 +1153196144100*A^2*B^8+14807038490244*A^2*B^7*C+70852371093504*A^2*B^6*C^2+ 170119495177632*A^2*B^5*C^3+225841932860544*A^2*B^4*C^4+170119495177632*A^2*B^3 *C^5+70852371093504*A^2*B^2*C^6+14807038490244*A^2*B*C^7+1153196144100*A^2*C^8+ 115319614410*A*B^9+2306392288200*A*B^8*C+14807038490244*A*B^7*C^2+ 46220101455528*A*B^6*C^3+79985684554776*A*B^5*C^4+79985684554776*A*B^4*C^5+ 46220101455528*A*B^3*C^6+14807038490244*A*B^2*C^7+2306392288200*A*B*C^8+ 115319614410*A*C^9+115319614410*B^9*C+1153196144100*B^8*C^2+4808827920897*B^7*C ^3+10793915908776*B^6*C^4+14068992958020*B^5*C^5+10793915908776*B^4*C^6+ 4808827920897*B^3*C^7+1153196144100*B^2*C^8+115319614410*B*C^9-55842438015*A^7* B-55842438015*A^7*C-446739504120*A^6*B^2-893479008240*A^6*B*C-446739504120*A^6* C^2-1377446804370*A^5*B^3-4318481873160*A^5*B^2*C-4318481873160*A^5*B*C^2-\ 1377446804370*A^5*C^3-2010327768540*A^4*B^4-9158159834460*A^4*B^3*C-\ 14295664131840*A^4*B^2*C^2-9158159834460*A^4*B*C^3-2010327768540*A^4*C^4-\ 1377446804370*A^3*B^5-9158159834460*A^3*B^4*C-20847843525600*A^3*B^3*C^2-\ 20847843525600*A^3*B^2*C^3-9158159834460*A^3*B*C^4-1377446804370*A^3*C^5-\ 446739504120*A^2*B^6-4318481873160*A^2*B^5*C-14295664131840*A^2*B^4*C^2-\ 20847843525600*A^2*B^3*C^3-14295664131840*A^2*B^2*C^4-4318481873160*A^2*B*C^5-\ 446739504120*A^2*C^6-55842438015*A*B^7-893479008240*A*B^6*C-4318481873160*A*B^5 *C^2-9158159834460*A*B^4*C^3-9158159834460*A*B^3*C^4-4318481873160*A*B^2*C^5-\ 893479008240*A*B*C^6-55842438015*A*C^7-55842438015*B^7*C-446739504120*B^6*C^2-\ 1377446804370*B^5*C^3-2010327768540*B^4*C^4-1377446804370*B^3*C^5-446739504120* B^2*C^6-55842438015*B*C^7+1844541888*A^5*B+1844541888*A^5*C+11067251328*A^4*B^2 +22134502656*A^4*B*C+11067251328*A^4*C^2+21212231712*A^3*B^3+70092591744*A^3*B^ 2*C+70092591744*A^3*B*C^2+21212231712*A^3*C^3+11067251328*A^2*B^4+70092591744*A ^2*B^3*C+118050680832*A^2*B^2*C^2+70092591744*A^2*B*C^3+11067251328*A^2*C^4+ 1844541888*A*B^5+22134502656*A*B^4*C+70092591744*A*B^3*C^2+70092591744*A*B^2*C^ 3+22134502656*A*B*C^4+1844541888*A*C^5+1844541888*B^5*C+11067251328*B^4*C^2+ 21212231712*B^3*C^3+11067251328*B^2*C^4+1844541888*B*C^5+23845772160*A^3*B+ 23845772160*A^3*C+95383088640*A^2*B^2+190766177280*A^2*B*C+95383088640*A^2*C^2+ 23845772160*A*B^3+190766177280*A*B^2*C+190766177280*A*B*C^2+23845772160*A*C^3+ 23845772160*B^3*C+95383088640*B^2*C^2+23845772160*B*C^3+96321757952*A*B+ 96321757952*A*C+96321757952*B*C+56313241600)/(A+B+C) Hence the, 10, -th moment about the mean is 26 26 25 2 3 A B C (35532270 A B + 35532270 A C + 959371290 A B 25 25 2 24 3 + 1918742580 A B C + 959371290 A C + 12354928947 A B 24 2 24 2 24 3 + 35778910788 A B C + 35778910788 A B C + 12354928947 A C 23 4 23 3 23 2 2 + 101009444175 A B + 373293649251 A B C + 544568410152 A B C 23 3 23 4 22 5 + 373293649251 A B C + 101009444175 A C + 588675830040 A B 22 4 22 3 2 + 2595590383644 A B C + 4866621605172 A B C 22 2 3 22 4 22 5 + 4866621605172 A B C + 2595590383644 A B C + 588675830040 A C 21 6 21 5 + 2603193448320 A B + 13143268159956 A B C 21 4 2 21 3 3 + 29627845821864 A B C + 38175542220456 A B C 21 2 4 21 5 + 29627845821864 A B C + 13143268159956 A B C 21 6 20 7 20 6 + 2603193448320 A C + 9077962435272 A B + 51101128434744 A B C 20 5 2 20 4 3 + 133122752806428 A B C + 207129038412618 A B C 20 3 4 20 2 5 + 207129038412618 A B C + 133122752806428 A B C 20 6 20 7 19 8 + 51101128434744 A B C + 9077962435272 A C + 25613110361592 A B 19 7 19 6 2 + 157939948886256 A B C + 463875694476516 A B C 19 5 3 19 4 4 + 838904628762270 A B C + 1014711545620836 A B C 19 3 5 19 2 6 + 838904628762270 A B C + 463875694476516 A B C 19 7 19 8 18 9 + 157939948886256 A B C + 25613110361592 A C + 59521207792230 A B 18 8 18 7 2 + 397539381189654 A B C + 1295794534417308 A B C 18 6 3 18 5 4 + 2658717914274324 A B C + 3746781504698400 A B C 18 4 5 18 3 6 + 3746781504698400 A B C + 2658717914274324 A B C 18 2 7 18 8 + 1295794534417308 A B C + 397539381189654 A B C 18 9 17 10 + 59521207792230 A C + 115373894719410 A B 17 9 17 8 2 + 829189989177516 A B C + 2969449074933678 A B C 17 7 3 17 6 4 + 6801068746009368 A B C + 10898687687059116 A B C 17 5 5 17 4 6 + 12706503843050640 A B C + 10898687687059116 A B C 17 3 7 17 2 8 + 6801068746009368 A B C + 2969449074933678 A B C 17 9 17 10 + 829189989177516 A B C + 115373894719410 A C 16 11 16 10 + 188215046961237 A B + 1451529482153220 A B C 16 9 2 16 8 3 + 5673964373944200 A B C + 14340291979046139 A B C 16 7 4 16 6 5 + 25700690373699294 A B C + 34106829915574836 A B C 16 5 6 16 4 7 + 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21526715825664*A^5*B^5*C-23122010214912*A^5*B^4*C^2-65024369077248*A^5*B^3*C^3-\ 23122010214912*A^5*B^2*C^4+21526715825664*A^5*B*C^5+12136541501952*A^5*C^6+ 8091027667968*A^4*B^7+18030475427328*A^4*B^6*C-23122010214912*A^4*B^5*C^2-\ 102881214105600*A^4*B^4*C^3-102881214105600*A^4*B^3*C^4-23122010214912*A^4*B^2* C^5+18030475427328*A^4*B*C^6+8091027667968*A^4*C^7+3467583286272*A^3*B^8+ 10460064082944*A^3*B^7*C-6591281227776*A^3*B^6*C^2-65024369077248*A^3*B^5*C^3-\ 102881214105600*A^3*B^4*C^4-65024369077248*A^3*B^3*C^5-6591281227776*A^3*B^2*C^ 6+10460064082944*A^3*B*C^7+3467583286272*A^3*C^8+866895821568*A^2*B^9+ 4020608573184*A^2*B^8*C+4738072829952*A^2*B^7*C^2-6591281227776*A^2*B^6*C^3-\ 23122010214912*A^2*B^5*C^4-23122010214912*A^2*B^4*C^5-6591281227776*A^2*B^3*C^6 +4738072829952*A^2*B^2*C^7+4020608573184*A^2*B*C^8+866895821568*A^2*C^9+ 96321757952*A*B^10+923983762688*A*B^9*C+4020608573184*A*B^8*C^2+10460064082944* A*B^7*C^3+18030475427328*A*B^6*C^4+21526715825664*A*B^5*C^5+18030475427328*A*B^ 4*C^6+10460064082944*A*B^3*C^7+4020608573184*A*B^2*C^8+923983762688*A*B*C^9+ 96321757952*A*C^10+96321757952*B^10*C+866895821568*B^9*C^2+3467583286272*B^8*C^ 3+8091027667968*B^7*C^4+12136541501952*B^6*C^5+12136541501952*B^5*C^6+ 8091027667968*B^4*C^7+3467583286272*B^3*C^8+866895821568*B^2*C^9+96321757952*B* C^10+56313241600*A^9+506819174400*A^8*B+506819174400*A^8*C+2027276697600*A^7*B^ 2+4054553395200*A^7*B*C+2027276697600*A^7*C^2+4730312294400*A^6*B^3+ 14190936883200*A^6*B^2*C+14190936883200*A^6*B*C^2+4730312294400*A^6*C^3+ 7095468441600*A^5*B^4+28381873766400*A^5*B^3*C+42572810649600*A^5*B^2*C^2+ 28381873766400*A^5*B*C^3+7095468441600*A^5*C^4+7095468441600*A^4*B^5+ 35477342208000*A^4*B^4*C+70954684416000*A^4*B^3*C^2+70954684416000*A^4*B^2*C^3+ 35477342208000*A^4*B*C^4+7095468441600*A^4*C^5+4730312294400*A^3*B^6+ 28381873766400*A^3*B^5*C+70954684416000*A^3*B^4*C^2+94606245888000*A^3*B^3*C^3+ 70954684416000*A^3*B^2*C^4+28381873766400*A^3*B*C^5+4730312294400*A^3*C^6+ 2027276697600*A^2*B^7+14190936883200*A^2*B^6*C+42572810649600*A^2*B^5*C^2+ 70954684416000*A^2*B^4*C^3+70954684416000*A^2*B^3*C^4+42572810649600*A^2*B^2*C^ 5+14190936883200*A^2*B*C^6+2027276697600*A^2*C^7+506819174400*A*B^8+ 4054553395200*A*B^7*C+14190936883200*A*B^6*C^2+28381873766400*A*B^5*C^3+ 35477342208000*A*B^4*C^4+28381873766400*A*B^3*C^5+14190936883200*A*B^2*C^6+ 4054553395200*A*B*C^7+506819174400*A*C^8+56313241600*B^9+506819174400*B^8*C+ 2027276697600*B^7*C^2+4730312294400*B^6*C^3+7095468441600*B^5*C^4+7095468441600 *B^4*C^5+4730312294400*B^3*C^6+2027276697600*B^2*C^7+506819174400*B*C^8+ 56313241600*C^9)/(A+B+C)^10 Hence the scaled, 10, -th moment about the mean is 26 26 25 2 25 (35532270 A B + 35532270 A C + 959371290 A B + 1918742580 A B C 25 2 24 3 24 2 + 959371290 A C + 12354928947 A B + 35778910788 A B C 24 2 24 3 23 4 + 35778910788 A B C + 12354928947 A C + 101009444175 A B 23 3 23 2 2 23 3 + 373293649251 A B C + 544568410152 A B C + 373293649251 A B C 23 4 22 5 22 4 + 101009444175 A C + 588675830040 A B + 2595590383644 A B C 22 3 2 22 2 3 + 4866621605172 A B C + 4866621605172 A B C 22 4 22 5 21 6 + 2595590383644 A B C + 588675830040 A C + 2603193448320 A B 21 5 21 4 2 + 13143268159956 A B C + 29627845821864 A B C 21 3 3 21 2 4 + 38175542220456 A B C + 29627845821864 A B C 21 5 21 6 20 7 + 13143268159956 A B C + 2603193448320 A C + 9077962435272 A B 20 6 20 5 2 + 51101128434744 A B C + 133122752806428 A B C 20 4 3 20 3 4 + 207129038412618 A B C + 207129038412618 A B C 20 2 5 20 6 + 133122752806428 A B C + 51101128434744 A B C 20 7 19 8 19 7 + 9077962435272 A C + 25613110361592 A B + 157939948886256 A B C 19 6 2 19 5 3 + 463875694476516 A B C + 838904628762270 A B C 19 4 4 19 3 5 + 1014711545620836 A B C + 838904628762270 A B C 19 2 6 19 7 + 463875694476516 A B C + 157939948886256 A B C 19 8 18 9 18 8 + 25613110361592 A C + 59521207792230 A B + 397539381189654 A B C 18 7 2 18 6 3 + 1295794534417308 A B C + 2658717914274324 A B C 18 5 4 18 4 5 + 3746781504698400 A B C + 3746781504698400 A B C 18 3 6 18 2 7 + 2658717914274324 A B C + 1295794534417308 A B C 18 8 18 9 + 397539381189654 A B C + 59521207792230 A C 17 10 17 9 + 115373894719410 A B + 829189989177516 A B C 17 8 2 17 7 3 + 2969449074933678 A B C + 6801068746009368 A B C 17 6 4 17 5 5 + 10898687687059116 A B C + 12706503843050640 A B C 17 4 6 17 3 7 + 10898687687059116 A B C + 6801068746009368 A B C 17 2 8 17 9 + 2969449074933678 A B C + 829189989177516 A B C 17 10 16 11 + 115373894719410 A C + 188215046961237 A B 16 10 16 9 2 + 1451529482153220 A B C + 5673964373944200 A B C 16 8 3 16 7 4 + 14340291979046139 A B C + 25700690373699294 A B C 16 6 5 16 5 6 + 34106829915574836 A B C + 34106829915574836 A B C 16 4 7 16 3 8 + 25700690373699294 A B C + 14340291979046139 A B C 16 2 9 16 10 + 5673964373944200 A B C + 1451529482153220 A B C 16 11 15 12 + 188215046961237 A C + 259999850944377 A B 15 11 15 10 2 + 2152070142932133 A B C + 9143007946280748 A B C 15 9 3 15 8 4 + 25288019663483655 A B C + 50101402524173613 A B C 15 7 5 15 6 6 + 74451174716161962 A B C + 84773708402358024 A B C 15 5 7 15 4 8 + 74451174716161962 A B C + 50101402524173613 A B C 15 3 9 15 2 10 + 25288019663483655 A B C + 9143007946280748 A B C 15 11 15 12 + 2152070142932133 A B C + 259999850944377 A C 14 13 14 12 + 305302729030200 A B + 2719150147910880 A B C 14 11 2 14 10 3 + 12519085846280904 A B C + 37670599264708800 A B C 14 9 4 14 8 5 + 81836661866034168 A B C + 134573173030794888 A B C 14 7 6 14 6 7 + 171610128361043928 A B C + 171610128361043928 A B C 14 5 8 14 4 9 + 134573173030794888 A B C + 81836661866034168 A B C 14 3 10 14 2 11 + 37670599264708800 A B C + 12519085846280904 A B C 14 12 14 13 + 2719150147910880 A B C + 305302729030200 A C 13 14 13 13 + 305302729030200 A B + 2938465965935016 A B C 13 12 2 13 11 3 + 14633279318716632 A B C + 47723003483787576 A B C 13 10 4 13 9 5 + 113012642250998280 A B C + 203811106069060560 A B C 13 8 6 13 7 7 + 287613076828812192 A B C + 322183451217548304 A B C 13 6 8 13 5 9 + 287613076828812192 A B C + 203811106069060560 A B C 13 4 10 13 3 11 + 113012642250998280 A B C + 47723003483787576 A B C 13 2 12 13 13 + 14633279318716632 A B C + 2938465965935016 A B C 13 14 12 15 + 305302729030200 A C + 259999850944377 A B 12 14 12 13 2 + 2719150147910880 A B C + 14633279318716632 A B C 12 12 3 12 11 4 + 51615856369873152 A B C + 132645851146389672 A B C 12 10 5 12 9 6 + 260625809491935552 A B C + 403704282412748016 A B C 12 8 7 12 7 8 + 500882306618495574 A B C + 500882306618495574 A B C 12 6 9 12 5 10 + 403704282412748016 A B C + 260625809491935552 A B C 12 4 11 12 3 12 + 132645851146389672 A B C + 51615856369873152 A B C 12 2 13 12 14 + 14633279318716632 A B C + 2719150147910880 A B C 12 15 11 16 + 259999850944377 A C + 188215046961237 A B 11 15 11 14 2 + 2152070142932133 A B C + 12519085846280904 A B C 11 13 3 11 12 4 + 47723003483787576 A B C + 132645851146389672 A B C 11 11 5 11 10 6 + 282730945692889512 A B C + 477758311425682560 A B C 11 9 7 11 8 8 + 649846242477634878 A B C + 718681596663859452 A B C 11 7 9 11 6 10 + 649846242477634878 A B C + 477758311425682560 A B C 11 5 11 11 4 12 + 282730945692889512 A B C + 132645851146389672 A B C 11 3 13 11 2 14 + 47723003483787576 A B C + 12519085846280904 A B C 11 15 11 16 + 2152070142932133 A B C + 188215046961237 A C 10 17 10 16 + 115373894719410 A B + 1451529482153220 A B C 10 15 2 10 14 3 + 9143007946280748 A B C + 37670599264708800 A B C 10 13 4 10 12 5 + 113012642250998280 A B C + 260625809491935552 A B C 10 11 6 10 10 7 + 477758311425682560 A B C + 707883758512298784 A B C 10 9 8 10 8 9 + 861502360237688052 A B C + 861502360237688052 A B C 10 7 10 10 6 11 + 707883758512298784 A B C + 477758311425682560 A B C 10 5 12 10 4 13 + 260625809491935552 A B C + 113012642250998280 A B C 10 3 14 10 2 15 + 37670599264708800 A B C + 9143007946280748 A B C 10 16 10 17 + 1451529482153220 A B C + 115373894719410 A C 9 18 9 17 + 59521207792230 A B + 829189989177516 A B C 9 16 2 9 15 3 + 5673964373944200 A B C + 25288019663483655 A B C 9 14 4 9 13 5 + 81836661866034168 A B C + 203811106069060560 A B C 9 12 6 9 11 7 + 403704282412748016 A B C + 649846242477634878 A B C 9 10 8 9 9 9 + 861502360237688052 A B C + 942279786934699356 A B C 9 8 10 9 7 11 + 861502360237688052 A B C + 649846242477634878 A B C 9 6 12 9 5 13 + 403704282412748016 A B C + 203811106069060560 A B C 9 4 14 9 3 15 + 81836661866034168 A B C + 25288019663483655 A B C 9 2 16 9 17 + 5673964373944200 A B C + 829189989177516 A B C 9 18 8 19 8 18 + 59521207792230 A C + 25613110361592 A B + 397539381189654 A B C 8 17 2 8 16 3 + 2969449074933678 A B C + 14340291979046139 A B C 8 15 4 8 14 5 + 50101402524173613 A B C + 134573173030794888 A B C 8 13 6 8 12 7 + 287613076828812192 A B C + 500882306618495574 A B C 8 11 8 8 10 9 + 718681596663859452 A B C + 861502360237688052 A B C 8 9 10 8 8 11 + 861502360237688052 A B C + 718681596663859452 A B C 8 7 12 8 6 13 + 500882306618495574 A B C + 287613076828812192 A B C 8 5 14 8 4 15 + 134573173030794888 A B C + 50101402524173613 A B C 8 3 16 8 2 17 + 14340291979046139 A B C + 2969449074933678 A B C 8 18 8 19 7 20 + 397539381189654 A B C + 25613110361592 A C + 9077962435272 A B 7 19 7 18 2 + 157939948886256 A B C + 1295794534417308 A B C 7 17 3 7 16 4 + 6801068746009368 A B C + 25700690373699294 A B C 7 15 5 7 14 6 + 74451174716161962 A B C + 171610128361043928 A B C 7 13 7 7 12 8 + 322183451217548304 A B C + 500882306618495574 A B C 7 11 9 7 10 10 + 649846242477634878 A B C + 707883758512298784 A B C 7 9 11 7 8 12 + 649846242477634878 A B C + 500882306618495574 A B C 7 7 13 7 6 14 + 322183451217548304 A B C + 171610128361043928 A B C 7 5 15 7 4 16 + 74451174716161962 A B C + 25700690373699294 A B C 7 3 17 7 2 18 + 6801068746009368 A B C + 1295794534417308 A B C 7 19 7 20 6 21 + 157939948886256 A B C + 9077962435272 A C + 2603193448320 A B 6 20 6 19 2 + 51101128434744 A B C + 463875694476516 A B C 6 18 3 6 17 4 + 2658717914274324 A B C + 10898687687059116 A B C 6 16 5 6 15 6 + 34106829915574836 A B C + 84773708402358024 A B C 6 14 7 6 13 8 + 171610128361043928 A B C + 287613076828812192 A B C 6 12 9 6 11 10 + 403704282412748016 A B C + 477758311425682560 A B C 6 10 11 6 9 12 + 477758311425682560 A B C + 403704282412748016 A B C 6 8 13 6 7 14 + 287613076828812192 A B C + 171610128361043928 A B C 6 6 15 6 5 16 + 84773708402358024 A B C + 34106829915574836 A B C 6 4 17 6 3 18 + 10898687687059116 A B C + 2658717914274324 A B C 6 2 19 6 20 + 463875694476516 A B C + 51101128434744 A B C 6 21 5 22 5 21 + 2603193448320 A C + 588675830040 A B + 13143268159956 A B C 5 20 2 5 19 3 + 133122752806428 A B C + 838904628762270 A B C 5 18 4 5 17 5 + 3746781504698400 A B C + 12706503843050640 A B C 5 16 6 5 15 7 + 34106829915574836 A B C + 74451174716161962 A B C 5 14 8 5 13 9 + 134573173030794888 A B C + 203811106069060560 A B C 5 12 10 5 11 11 + 260625809491935552 A B C + 282730945692889512 A B C 5 10 12 5 9 13 + 260625809491935552 A B C + 203811106069060560 A B C 5 8 14 5 7 15 + 134573173030794888 A B C + 74451174716161962 A B C 5 6 16 5 5 17 + 34106829915574836 A B C + 12706503843050640 A B C 5 4 18 5 3 19 + 3746781504698400 A B C + 838904628762270 A B C 5 2 20 5 21 + 133122752806428 A B C + 13143268159956 A B C 5 22 4 23 4 22 + 588675830040 A C + 101009444175 A B + 2595590383644 A B C 4 21 2 4 20 3 + 29627845821864 A B C + 207129038412618 A B C 4 19 4 4 18 5 + 1014711545620836 A B C + 3746781504698400 A B C 4 17 6 4 16 7 + 10898687687059116 A B C + 25700690373699294 A B C 4 15 8 4 14 9 + 50101402524173613 A B C + 81836661866034168 A B C 4 13 10 4 12 11 + 113012642250998280 A B C + 132645851146389672 A B C 4 11 12 4 10 13 + 132645851146389672 A B C + 113012642250998280 A B C 4 9 14 4 8 15 + 81836661866034168 A B C + 50101402524173613 A B C 4 7 16 4 6 17 + 25700690373699294 A B C + 10898687687059116 A B C 4 5 18 4 4 19 + 3746781504698400 A B C + 1014711545620836 A B C 4 3 20 4 2 21 + 207129038412618 A B C + 29627845821864 A B C 4 22 4 23 3 24 + 2595590383644 A B C + 101009444175 A C + 12354928947 A B 3 23 3 22 2 + 373293649251 A B C + 4866621605172 A B C 3 21 3 3 20 4 + 38175542220456 A B C + 207129038412618 A B C 3 19 5 3 18 6 + 838904628762270 A B C + 2658717914274324 A B C 3 17 7 3 16 8 + 6801068746009368 A B C + 14340291979046139 A B C 3 15 9 3 14 10 + 25288019663483655 A B C + 37670599264708800 A B C 3 13 11 3 12 12 + 47723003483787576 A B C + 51615856369873152 A B C 3 11 13 3 10 14 + 47723003483787576 A B C + 37670599264708800 A B C 3 9 15 3 8 16 + 25288019663483655 A B C + 14340291979046139 A B C 3 7 17 3 6 18 + 6801068746009368 A B C + 2658717914274324 A B C 3 5 19 3 4 20 + 838904628762270 A B C + 207129038412618 A B C 3 3 21 3 2 22 + 38175542220456 A B C + 4866621605172 A B C 3 23 3 24 2 25 + 373293649251 A B C + 12354928947 A C + 959371290 A B 2 24 2 23 2 2 22 3 + 35778910788 A B C + 544568410152 A B C + 4866621605172 A B C 2 21 4 2 20 5 + 29627845821864 A B C + 133122752806428 A B C 2 19 6 2 18 7 + 463875694476516 A B C + 1295794534417308 A B C 2 17 8 2 16 9 + 2969449074933678 A B C + 5673964373944200 A B C 2 15 10 2 14 11 + 9143007946280748 A B C + 12519085846280904 A B C 2 13 12 2 12 13 + 14633279318716632 A B C + 14633279318716632 A B C 2 11 14 2 10 15 + 12519085846280904 A B C + 9143007946280748 A B C 2 9 16 2 8 17 + 5673964373944200 A B C + 2969449074933678 A B C 2 7 18 2 6 19 + 1295794534417308 A B C + 463875694476516 A B C 2 5 20 2 4 21 + 133122752806428 A B C + 29627845821864 A B C 2 3 22 2 2 23 2 24 + 4866621605172 A B C + 544568410152 A B C + 35778910788 A B C 2 25 26 25 + 959371290 A C + 35532270 A B + 1918742580 A B C 24 2 23 3 22 4 + 35778910788 A B C + 373293649251 A B C + 2595590383644 A B C 21 5 20 6 + 13143268159956 A B C + 51101128434744 A B C 19 7 18 8 + 157939948886256 A B C + 397539381189654 A B C 17 9 16 10 + 829189989177516 A B C + 1451529482153220 A B C 15 11 14 12 + 2152070142932133 A B C + 2719150147910880 A B C 13 13 12 14 + 2938465965935016 A B C + 2719150147910880 A B C 11 15 10 16 + 2152070142932133 A B C + 1451529482153220 A B C 9 17 8 18 + 829189989177516 A B C + 397539381189654 A B C 7 19 6 20 + 157939948886256 A B C + 51101128434744 A B C 5 21 4 22 3 23 + 13143268159956 A B C + 2595590383644 A B C + 373293649251 A B C 2 24 25 26 + 35778910788 A B C + 1918742580 A B C + 35532270 A C 26 25 2 24 3 + 35532270 B C + 959371290 B C + 12354928947 B C 23 4 22 5 21 6 + 101009444175 B C + 588675830040 B C + 2603193448320 B C 20 7 19 8 18 9 + 9077962435272 B C + 25613110361592 B C + 59521207792230 B C 17 10 16 11 + 115373894719410 B C + 188215046961237 B C 15 12 14 13 + 259999850944377 B C + 305302729030200 B C 13 14 12 15 + 305302729030200 B C + 259999850944377 B C 11 16 10 17 + 188215046961237 B C + 115373894719410 B C 9 18 8 19 7 20 + 59521207792230 B C + 25613110361592 B C + 9077962435272 B C 6 21 5 22 4 23 + 2603193448320 B C + 588675830040 B C + 101009444175 B C 3 24 2 25 26 + 12354928947 B C + 959371290 B C + 35532270 B C 24 24 23 2 - 1052080407 A B - 1052080407 A C - 26302010175 A B 23 23 2 22 3 - 52604020350 A B C - 26302010175 A C - 312162757200 A B 22 2 22 2 22 3 - 903028410900 A B C - 903028410900 A B C - 312162757200 A C 21 4 21 3 21 2 2 - 2340173543400 A B - 8628038603100 A B C - 12575730119400 A B C 21 3 21 4 20 5 - 8628038603100 A B C - 2340173543400 A C - 12437647644366 A B 20 4 20 3 2 - 54647537210658 A B C - 102275254984140 A B C 20 2 3 20 4 - 102275254984140 A B C - 54647537210658 A B C 20 5 19 6 19 5 - 12437647644366 A C - 49862626911846 A B - 250685575308504 A B C 19 4 2 19 3 3 - 563763032150598 A B C - 725880167507880 A B C 19 2 4 19 5 - 563763032150598 A B C - 250685575308504 A B C 19 6 18 7 - 49862626911846 A C - 156626561024010 A B 18 6 18 5 2 - 877976554587300 A B C - 2282260440198060 A B C 18 4 3 18 3 4 - 3548027952686700 A B C - 3548027952686700 A B C 18 2 5 18 6 - 2282260440198060 A B C - 877976554587300 A B C 18 7 17 8 - 156626561024010 A C - 395261750394630 A B 17 7 17 6 2 - 2429963587643130 A B C - 7130008833786720 A B C 17 5 3 17 4 4 - 12892236251572440 A B C - 15593858510068440 A B C 17 3 5 17 2 6 - 12892236251572440 A B C - 7130008833786720 A B C 17 7 17 8 - 2429963587643130 A B C - 395261750394630 A C 16 9 16 8 - 815207818420737 A B - 5442837169241115 A B C 16 7 2 16 6 3 - 17764054173225360 A B C - 36493070032597500 A B C 16 5 4 16 4 5 - 51460227401619834 A B C - 51460227401619834 A B C 16 3 6 16 2 7 - 36493070032597500 A B C - 17764054173225360 A B C 16 8 16 9 - 5442837169241115 A B C - 815207818420737 A C 15 10 15 9 - 1390411979599293 A B - 10034651953805976 A B C 15 8 2 15 7 3 - 36097561969246575 A B C - 82937004582821760 A B C 15 6 4 15 5 5 - 133143422101326630 A B C - 155319479027089524 A B C 15 4 6 15 3 7 - 133143422101326630 A B C - 82937004582821760 A B C 15 2 8 15 9 - 36097561969246575 A B C - 10034651953805976 A B C 15 10 14 11 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724325766931872*A^4*B^8*C^3-1722732256538496*A^4*B^7*C^4-2552806002563712*A^4*B ^6*C^5-2552806002563712*A^4*B^5*C^6-1722732256538496*A^4*B^4*C^7-\ 724325766931872*A^4*B^3*C^8-154433830251456*A^4*B^2*C^9-7387778357568*A^4*B*C^ 10+755339903136*A^4*C^11+187221001632*A^3*B^12-435947642592*A^3*B^11*C-\ 33535713740928*A^3*B^10*C^2-225597409074912*A^3*B^9*C^3-724325766931872*A^3*B^8 *C^4-1385859158784576*A^3*B^7*C^5-1708436511661824*A^3*B^6*C^6-1385859158784576 *A^3*B^5*C^7-724325766931872*A^3*B^4*C^8-225597409074912*A^3*B^3*C^9-\ 33535713740928*A^3*B^2*C^10-435947642592*A^3*B*C^11+187221001632*A^3*C^12+ 27668128320*A^2*B^13+273016075584*A^2*B^12*C-2382575091456*A^2*B^11*C^2-\ 33535713740928*A^2*B^10*C^3-154433830251456*A^2*B^9*C^4-386644496107968*A^2*B^8 *C^5-598427666266752*A^2*B^7*C^6-598427666266752*A^2*B^6*C^7-386644496107968*A^ 2*B^5*C^8-154433830251456*A^2*B^4*C^9-33535713740928*A^2*B^3*C^10-2382575091456 *A^2*B^2*C^11+273016075584*A^2*B*C^12+27668128320*A^2*C^13+1844541888*A*B^14+ 55336256640*A*B^13*C+273016075584*A*B^12*C^2-435947642592*A*B^11*C^3-\ 7387778357568*A*B^10*C^4-27405690251520*A*B^9*C^5-55031597272512*A*B^8*C^6-\ 68716426750272*A*B^7*C^7-55031597272512*A*B^6*C^8-27405690251520*A*B^5*C^9-\ 7387778357568*A*B^4*C^10-435947642592*A*B^3*C^11+273016075584*A*B^2*C^12+ 55336256640*A*B*C^13+1844541888*A*C^14+1844541888*B^14*C+27668128320*B^13*C^2+ 187221001632*B^12*C^3+755339903136*B^11*C^4+2027151534912*B^10*C^5+ 3823735333824*B^9*C^6+5218209001152*B^8*C^7+5218209001152*B^7*C^8+3823735333824 *B^6*C^9+2027151534912*B^5*C^10+755339903136*B^4*C^11+187221001632*B^3*C^12+ 27668128320*B^2*C^13+1844541888*B*C^14+23845772160*A^12*B+23845772160*A^12*C+ 309995038080*A^11*B^2+619990076160*A^11*B*C+309995038080*A^11*C^2+1740741367680 *A^10*B^3+3888775296000*A^10*B^2*C+3888775296000*A^10*B*C^2+1740741367680*A^10* C^3+5651448001920*A^9*B^4+12081276184320*A^9*B^3*C+11389032506880*A^9*B^2*C^2+ 12081276184320*A^9*B*C^3+5651448001920*A^9*C^4+11875194535680*A^8*B^5+ 23207848917120*A^8*B^4*C-1404087511680*A^8*B^3*C^2-1404087511680*A^8*B^2*C^3+ 23207848917120*A^8*B*C^4+11875194535680*A^8*C^5+17025881322240*A^7*B^6+ 31679010639360*A^7*B^5*C-66853360869120*A^7*B^4*C^2-163012980372480*A^7*B^3*C^3 -66853360869120*A^7*B^2*C^4+31679010639360*A^7*B*C^5+17025881322240*A^7*C^6+ 17025881322240*A^6*B^7+34519613829120*A^6*B^6*C-145405449745920*A^6*B^5*C^2-\ 489633249127680*A^6*B^4*C^3-489633249127680*A^6*B^3*C^4-145405449745920*A^6*B^2 *C^5+34519613829120*A^6*B*C^6+17025881322240*A^6*C^7+11875194535680*A^5*B^8+ 31679010639360*A^5*B^7*C-145405449745920*A^5*B^6*C^2-676729782167040*A^5*B^5*C^ 3-1023041032634880*A^5*B^4*C^4-676729782167040*A^5*B^3*C^5-145405449745920*A^5* B^2*C^6+31679010639360*A^5*B*C^7+11875194535680*A^5*C^8+5651448001920*A^4*B^9+ 23207848917120*A^4*B^8*C-66853360869120*A^4*B^7*C^2-489633249127680*A^4*B^6*C^3 -1023041032634880*A^4*B^5*C^4-1023041032634880*A^4*B^4*C^5-489633249127680*A^4* B^3*C^6-66853360869120*A^4*B^2*C^7+23207848917120*A^4*B*C^8+5651448001920*A^4*C ^9+1740741367680*A^3*B^10+12081276184320*A^3*B^9*C-1404087511680*A^3*B^8*C^2-\ 163012980372480*A^3*B^7*C^3-489633249127680*A^3*B^6*C^4-676729782167040*A^3*B^5 *C^5-489633249127680*A^3*B^4*C^6-163012980372480*A^3*B^3*C^7-1404087511680*A^3* B^2*C^8+12081276184320*A^3*B*C^9+1740741367680*A^3*C^10+309995038080*A^2*B^11+ 3888775296000*A^2*B^10*C+11389032506880*A^2*B^9*C^2-1404087511680*A^2*B^8*C^3-\ 66853360869120*A^2*B^7*C^4-145405449745920*A^2*B^6*C^5-145405449745920*A^2*B^5* C^6-66853360869120*A^2*B^4*C^7-1404087511680*A^2*B^3*C^8+11389032506880*A^2*B^2 *C^9+3888775296000*A^2*B*C^10+309995038080*A^2*C^11+23845772160*A*B^12+ 619990076160*A*B^11*C+3888775296000*A*B^10*C^2+12081276184320*A*B^9*C^3+ 23207848917120*A*B^8*C^4+31679010639360*A*B^7*C^5+34519613829120*A*B^6*C^6+ 31679010639360*A*B^5*C^7+23207848917120*A*B^4*C^8+12081276184320*A*B^3*C^9+ 3888775296000*A*B^2*C^10+619990076160*A*B*C^11+23845772160*A*C^12+23845772160*B ^12*C+309995038080*B^11*C^2+1740741367680*B^10*C^3+5651448001920*B^9*C^4+ 11875194535680*B^8*C^5+17025881322240*B^7*C^6+17025881322240*B^6*C^7+ 11875194535680*B^5*C^8+5651448001920*B^4*C^9+1740741367680*B^3*C^10+ 309995038080*B^2*C^11+23845772160*B*C^12+96321757952*A^10*B+96321757952*A^10*C+ 866895821568*A^9*B^2+923983762688*A^9*B*C+866895821568*A^9*C^2+3467583286272*A^ 8*B^3+4020608573184*A^8*B^2*C+4020608573184*A^8*B*C^2+3467583286272*A^8*C^3+ 8091027667968*A^7*B^4+10460064082944*A^7*B^3*C+4738072829952*A^7*B^2*C^2+ 10460064082944*A^7*B*C^3+8091027667968*A^7*C^4+12136541501952*A^6*B^5+ 18030475427328*A^6*B^4*C-6591281227776*A^6*B^3*C^2-6591281227776*A^6*B^2*C^3+ 18030475427328*A^6*B*C^4+12136541501952*A^6*C^5+12136541501952*A^5*B^6+ 21526715825664*A^5*B^5*C-23122010214912*A^5*B^4*C^2-65024369077248*A^5*B^3*C^3-\ 23122010214912*A^5*B^2*C^4+21526715825664*A^5*B*C^5+12136541501952*A^5*C^6+ 8091027667968*A^4*B^7+18030475427328*A^4*B^6*C-23122010214912*A^4*B^5*C^2-\ 102881214105600*A^4*B^4*C^3-102881214105600*A^4*B^3*C^4-23122010214912*A^4*B^2* C^5+18030475427328*A^4*B*C^6+8091027667968*A^4*C^7+3467583286272*A^3*B^8+ 10460064082944*A^3*B^7*C-6591281227776*A^3*B^6*C^2-65024369077248*A^3*B^5*C^3-\ 102881214105600*A^3*B^4*C^4-65024369077248*A^3*B^3*C^5-6591281227776*A^3*B^2*C^ 6+10460064082944*A^3*B*C^7+3467583286272*A^3*C^8+866895821568*A^2*B^9+ 4020608573184*A^2*B^8*C+4738072829952*A^2*B^7*C^2-6591281227776*A^2*B^6*C^3-\ 23122010214912*A^2*B^5*C^4-23122010214912*A^2*B^4*C^5-6591281227776*A^2*B^3*C^6 +4738072829952*A^2*B^2*C^7+4020608573184*A^2*B*C^8+866895821568*A^2*C^9+ 96321757952*A*B^10+923983762688*A*B^9*C+4020608573184*A*B^8*C^2+10460064082944* A*B^7*C^3+18030475427328*A*B^6*C^4+21526715825664*A*B^5*C^5+18030475427328*A*B^ 4*C^6+10460064082944*A*B^3*C^7+4020608573184*A*B^2*C^8+923983762688*A*B*C^9+ 96321757952*A*C^10+96321757952*B^10*C+866895821568*B^9*C^2+3467583286272*B^8*C^ 3+8091027667968*B^7*C^4+12136541501952*B^6*C^5+12136541501952*B^5*C^6+ 8091027667968*B^4*C^7+3467583286272*B^3*C^8+866895821568*B^2*C^9+96321757952*B* C^10+56313241600*A^9+506819174400*A^8*B+506819174400*A^8*C+2027276697600*A^7*B^ 2+4054553395200*A^7*B*C+2027276697600*A^7*C^2+4730312294400*A^6*B^3+ 14190936883200*A^6*B^2*C+14190936883200*A^6*B*C^2+4730312294400*A^6*C^3+ 7095468441600*A^5*B^4+28381873766400*A^5*B^3*C+42572810649600*A^5*B^2*C^2+ 28381873766400*A^5*B*C^3+7095468441600*A^5*C^4+7095468441600*A^4*B^5+ 35477342208000*A^4*B^4*C+70954684416000*A^4*B^3*C^2+70954684416000*A^4*B^2*C^3+ 35477342208000*A^4*B*C^4+7095468441600*A^4*C^5+4730312294400*A^3*B^6+ 28381873766400*A^3*B^5*C+70954684416000*A^3*B^4*C^2+94606245888000*A^3*B^3*C^3+ 70954684416000*A^3*B^2*C^4+28381873766400*A^3*B*C^5+4730312294400*A^3*C^6+ 2027276697600*A^2*B^7+14190936883200*A^2*B^6*C+42572810649600*A^2*B^5*C^2+ 70954684416000*A^2*B^4*C^3+70954684416000*A^2*B^3*C^4+42572810649600*A^2*B^2*C^ 5+14190936883200*A^2*B*C^6+2027276697600*A^2*C^7+506819174400*A*B^8+ 4054553395200*A*B^7*C+14190936883200*A*B^6*C^2+28381873766400*A*B^5*C^3+ 35477342208000*A*B^4*C^4+28381873766400*A*B^3*C^5+14190936883200*A*B^2*C^6+ 4054553395200*A*B*C^7+506819174400*A*C^8+56313241600*B^9+506819174400*B^8*C+ 2027276697600*B^7*C^2+4730312294400*B^6*C^3+7095468441600*B^5*C^4+7095468441600 *B^4*C^5+4730312294400*B^3*C^6+2027276697600*B^2*C^7+506819174400*B*C^8+ 56313241600*C^9)/(A^2*B+A^2*C+A*B^2-3*A*B*C+A*C^2+B^2*C+B*C^2-A-B-C)^5 Note that if N is large and the starting capitals are x1*N,x2*N,x3*N, (where\ , of course, x1+x2+x3=1), then the limit of the, 10, -th scaled moment about the mean tends to 26 26 25 2 25 (438670 x1 x2 + 438670 x1 x3 + 11844090 x1 x2 + 23688180 x1 x2 x3 25 2 24 3 24 2 + 11844090 x1 x3 + 152529987 x1 x2 + 441714948 x1 x2 x3 24 2 24 3 23 4 + 441714948 x1 x2 x3 + 152529987 x1 x3 + 1247030175 x1 x2 23 3 23 2 2 + 4608563571 x1 x2 x3 + 6723066792 x1 x2 x3 23 3 23 4 22 5 + 4608563571 x1 x2 x3 + 1247030175 x1 x3 + 7267602840 x1 x2 22 4 22 3 2 + 32044325724 x1 x2 x3 + 60081748212 x1 x2 x3 22 2 3 22 4 22 5 + 60081748212 x1 x2 x3 + 32044325724 x1 x2 x3 + 7267602840 x1 x3 21 6 21 5 + 32138190720 x1 x2 + 162262569876 x1 x2 x3 21 4 2 21 3 3 + 365775874344 x1 x2 x3 + 471302990376 x1 x2 x3 21 2 4 21 5 + 365775874344 x1 x2 x3 + 162262569876 x1 x2 x3 21 6 20 7 20 6 + 32138190720 x1 x3 + 112073610312 x1 x2 + 630878128824 x1 x2 x3 20 5 2 20 4 3 + 1643490775388 x1 x2 x3 + 2557148622378 x1 x2 x3 20 3 4 20 2 5 + 2557148622378 x1 x2 x3 + 1643490775388 x1 x2 x3 20 6 20 7 19 8 + 630878128824 x1 x2 x3 + 112073610312 x1 x3 + 316211239032 x1 x2 19 7 19 6 2 + 1949875912176 x1 x2 x3 + 5726860425636 x1 x2 x3 19 5 3 19 4 4 + 10356847268670 x1 x2 x3 + 12527303032356 x1 x2 x3 19 3 5 19 2 6 + 10356847268670 x1 x2 x3 + 5726860425636 x1 x2 x3 19 7 19 8 + 1949875912176 x1 x2 x3 + 316211239032 x1 x3 18 9 18 8 + 734829725830 x1 x2 + 4907893594934 x1 x2 x3 18 7 2 18 6 3 + 15997463387868 x1 x2 x3 + 32823677954004 x1 x2 x3 18 5 4 18 4 5 + 46256561786400 x1 x2 x3 + 46256561786400 x1 x2 x3 18 3 6 18 2 7 + 32823677954004 x1 x2 x3 + 15997463387868 x1 x2 x3 18 8 18 9 + 4907893594934 x1 x2 x3 + 734829725830 x1 x3 17 10 17 9 + 1424369070610 x1 x2 + 10236913446636 x1 x2 x3 17 8 2 17 7 3 + 36659865122638 x1 x2 x3 + 83963811679128 x1 x2 x3 17 6 4 17 5 5 + 134551699840236 x1 x2 x3 + 156870417815440 x1 x2 x3 17 4 6 17 3 7 + 134551699840236 x1 x2 x3 + 83963811679128 x1 x2 x3 17 2 8 17 9 + 36659865122638 x1 x2 x3 + 10236913446636 x1 x2 x3 17 10 16 11 + 1424369070610 x1 x3 + 2323642555077 x1 x2 16 10 16 9 2 + 17920117063620 x1 x2 x3 + 70048942888200 x1 x2 x3 16 8 3 16 7 4 + 177040641716619 x1 x2 x3 + 317292473749374 x1 x2 x3 16 6 5 16 5 6 + 421071974266356 x1 x2 x3 + 421071974266356 x1 x2 x3 16 4 7 16 3 8 + 317292473749374 x1 x2 x3 + 177040641716619 x1 x2 x3 16 2 9 16 10 + 70048942888200 x1 x2 x3 + 17920117063620 x1 x2 x3 16 11 15 12 + 2323642555077 x1 x3 + 3209874703017 x1 x2 15 11 15 10 2 + 26568767196693 x1 x2 x3 + 112876641312108 x1 x2 x3 15 9 3 15 8 4 + 312197773623255 x1 x2 x3 + 618535833631773 x1 x2 x3 15 7 5 15 6 6 + 919150305137802 x1 x2 x3 + 1046588992621704 x1 x2 x3 15 5 7 15 4 8 + 919150305137802 x1 x2 x3 + 618535833631773 x1 x2 x3 15 3 9 15 2 10 + 312197773623255 x1 x2 x3 + 112876641312108 x1 x2 x3 15 11 15 12 + 26568767196693 x1 x2 x3 + 3209874703017 x1 x3 14 13 14 12 + 3769169494200 x1 x2 + 33569754912480 x1 x2 x3 14 11 2 14 10 3 + 154556615386184 x1 x2 x3 + 465069126724800 x1 x2 x3 14 9 4 14 8 5 + 1010329158839928 x1 x2 x3 + 1661397197911048 x1 x2 x3 14 7 6 14 6 7 + 2118643560012888 x1 x2 x3 + 2118643560012888 x1 x2 x3 14 5 8 14 4 9 + 1661397197911048 x1 x2 x3 + 1010329158839928 x1 x2 x3 14 3 10 14 2 11 + 465069126724800 x1 x2 x3 + 154556615386184 x1 x2 x3 14 12 14 13 + 33569754912480 x1 x2 x3 + 3769169494200 x1 x3 13 14 13 13 + 3769169494200 x1 x2 + 36277357604136 x1 x2 x3 13 12 2 13 11 3 + 180657769366872 x1 x2 x3 + 589172882515896 x1 x2 x3 13 10 4 13 9 5 + 1395217805567880 x1 x2 x3 + 2516186494679760 x1 x2 x3 13 8 6 13 7 7 + 3550778726281632 x1 x2 x3 + 3977573471821584 x1 x2 x3 13 6 8 13 5 9 + 3550778726281632 x1 x2 x3 + 2516186494679760 x1 x2 x3 13 4 10 13 3 11 + 1395217805567880 x1 x2 x3 + 589172882515896 x1 x2 x3 13 2 12 13 13 + 180657769366872 x1 x2 x3 + 36277357604136 x1 x2 x3 13 14 12 15 + 3769169494200 x1 x3 + 3209874703017 x1 x2 12 14 12 13 2 + 33569754912480 x1 x2 x3 + 180657769366872 x1 x2 x3 12 12 3 12 11 4 + 637232794689792 x1 x2 x3 + 1637603100572712 x1 x2 x3 12 10 5 12 9 6 + 3217602586320192 x1 x2 x3 + 4984003486577136 x1 x2 x3 12 8 7 12 7 8 + 6183732180475254 x1 x2 x3 + 6183732180475254 x1 x2 x3 12 6 9 12 5 10 + 4984003486577136 x1 x2 x3 + 3217602586320192 x1 x2 x3 12 4 11 12 3 12 + 1637603100572712 x1 x2 x3 + 637232794689792 x1 x2 x3 12 2 13 12 14 + 180657769366872 x1 x2 x3 + 33569754912480 x1 x2 x3 12 15 11 16 + 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6183732180475254*x1^8*x2^7*x3^12+3550778726281632*x1^8*x2^6*x3^13+ 1661397197911048*x1^8*x2^5*x3^14+618535833631773*x1^8*x2^4*x3^15+ 177040641716619*x1^8*x2^3*x3^16+36659865122638*x1^8*x2^2*x3^17+4907893594934*x1 ^8*x2*x3^18+316211239032*x1^8*x3^19+112073610312*x1^7*x2^20+1949875912176*x1^7* x2^19*x3+15997463387868*x1^7*x2^18*x3^2+83963811679128*x1^7*x2^17*x3^3+ 317292473749374*x1^7*x2^16*x3^4+919150305137802*x1^7*x2^15*x3^5+ 2118643560012888*x1^7*x2^14*x3^6+3977573471821584*x1^7*x2^13*x3^7+ 6183732180475254*x1^7*x2^12*x3^8+8022793117007838*x1^7*x2^11*x3^9+ 8739305660645664*x1^7*x2^10*x3^10+8022793117007838*x1^7*x2^9*x3^11+ 6183732180475254*x1^7*x2^8*x3^12+3977573471821584*x1^7*x2^7*x3^13+ 2118643560012888*x1^7*x2^6*x3^14+919150305137802*x1^7*x2^5*x3^15+ 317292473749374*x1^7*x2^4*x3^16+83963811679128*x1^7*x2^3*x3^17+15997463387868* x1^7*x2^2*x3^18+1949875912176*x1^7*x2*x3^19+112073610312*x1^7*x3^20+32138190720 *x1^6*x2^21+630878128824*x1^6*x2^20*x3+5726860425636*x1^6*x2^19*x3^2+ 32823677954004*x1^6*x2^18*x3^3+134551699840236*x1^6*x2^17*x3^4+421071974266356* x1^6*x2^16*x3^5+1046588992621704*x1^6*x2^15*x3^6+2118643560012888*x1^6*x2^14*x3 ^7+3550778726281632*x1^6*x2^13*x3^8+4984003486577136*x1^6*x2^12*x3^9+ 5898250758341760*x1^6*x2^11*x3^10+5898250758341760*x1^6*x2^10*x3^11+ 4984003486577136*x1^6*x2^9*x3^12+3550778726281632*x1^6*x2^8*x3^13+ 2118643560012888*x1^6*x2^7*x3^14+1046588992621704*x1^6*x2^6*x3^15+ 421071974266356*x1^6*x2^5*x3^16+134551699840236*x1^6*x2^4*x3^17+32823677954004* x1^6*x2^3*x3^18+5726860425636*x1^6*x2^2*x3^19+630878128824*x1^6*x2*x3^20+ 32138190720*x1^6*x3^21+7267602840*x1^5*x2^22+162262569876*x1^5*x2^21*x3+ 1643490775388*x1^5*x2^20*x3^2+10356847268670*x1^5*x2^19*x3^3+46256561786400*x1^ 5*x2^18*x3^4+156870417815440*x1^5*x2^17*x3^5+421071974266356*x1^5*x2^16*x3^6+ 919150305137802*x1^5*x2^15*x3^7+1661397197911048*x1^5*x2^14*x3^8+ 2516186494679760*x1^5*x2^13*x3^9+3217602586320192*x1^5*x2^12*x3^10+ 3490505502381352*x1^5*x2^11*x3^11+3217602586320192*x1^5*x2^10*x3^12+ 2516186494679760*x1^5*x2^9*x3^13+1661397197911048*x1^5*x2^8*x3^14+ 919150305137802*x1^5*x2^7*x3^15+421071974266356*x1^5*x2^6*x3^16+156870417815440 *x1^5*x2^5*x3^17+46256561786400*x1^5*x2^4*x3^18+10356847268670*x1^5*x2^3*x3^19+ 1643490775388*x1^5*x2^2*x3^20+162262569876*x1^5*x2*x3^21+7267602840*x1^5*x3^22+ 1247030175*x1^4*x2^23+32044325724*x1^4*x2^22*x3+365775874344*x1^4*x2^21*x3^2+ 2557148622378*x1^4*x2^20*x3^3+12527303032356*x1^4*x2^19*x3^4+46256561786400*x1^ 4*x2^18*x3^5+134551699840236*x1^4*x2^17*x3^6+317292473749374*x1^4*x2^16*x3^7+ 618535833631773*x1^4*x2^15*x3^8+1010329158839928*x1^4*x2^14*x3^9+ 1395217805567880*x1^4*x2^13*x3^10+1637603100572712*x1^4*x2^12*x3^11+ 1637603100572712*x1^4*x2^11*x3^12+1395217805567880*x1^4*x2^10*x3^13+ 1010329158839928*x1^4*x2^9*x3^14+618535833631773*x1^4*x2^8*x3^15+ 317292473749374*x1^4*x2^7*x3^16+134551699840236*x1^4*x2^6*x3^17+46256561786400* x1^4*x2^5*x3^18+12527303032356*x1^4*x2^4*x3^19+2557148622378*x1^4*x2^3*x3^20+ 365775874344*x1^4*x2^2*x3^21+32044325724*x1^4*x2*x3^22+1247030175*x1^4*x3^23+ 152529987*x1^3*x2^24+4608563571*x1^3*x2^23*x3+60081748212*x1^3*x2^22*x3^2+ 471302990376*x1^3*x2^21*x3^3+2557148622378*x1^3*x2^20*x3^4+10356847268670*x1^3* x2^19*x3^5+32823677954004*x1^3*x2^18*x3^6+83963811679128*x1^3*x2^17*x3^7+ 177040641716619*x1^3*x2^16*x3^8+312197773623255*x1^3*x2^15*x3^9+465069126724800 *x1^3*x2^14*x3^10+589172882515896*x1^3*x2^13*x3^11+637232794689792*x1^3*x2^12* x3^12+589172882515896*x1^3*x2^11*x3^13+465069126724800*x1^3*x2^10*x3^14+ 312197773623255*x1^3*x2^9*x3^15+177040641716619*x1^3*x2^8*x3^16+83963811679128* x1^3*x2^7*x3^17+32823677954004*x1^3*x2^6*x3^18+10356847268670*x1^3*x2^5*x3^19+ 2557148622378*x1^3*x2^4*x3^20+471302990376*x1^3*x2^3*x3^21+60081748212*x1^3*x2^ 2*x3^22+4608563571*x1^3*x2*x3^23+152529987*x1^3*x3^24+11844090*x1^2*x2^25+ 441714948*x1^2*x2^24*x3+6723066792*x1^2*x2^23*x3^2+60081748212*x1^2*x2^22*x3^3+ 365775874344*x1^2*x2^21*x3^4+1643490775388*x1^2*x2^20*x3^5+5726860425636*x1^2* x2^19*x3^6+15997463387868*x1^2*x2^18*x3^7+36659865122638*x1^2*x2^17*x3^8+ 70048942888200*x1^2*x2^16*x3^9+112876641312108*x1^2*x2^15*x3^10+154556615386184 *x1^2*x2^14*x3^11+180657769366872*x1^2*x2^13*x3^12+180657769366872*x1^2*x2^12* x3^13+154556615386184*x1^2*x2^11*x3^14+112876641312108*x1^2*x2^10*x3^15+ 70048942888200*x1^2*x2^9*x3^16+36659865122638*x1^2*x2^8*x3^17+15997463387868*x1 ^2*x2^7*x3^18+5726860425636*x1^2*x2^6*x3^19+1643490775388*x1^2*x2^5*x3^20+ 365775874344*x1^2*x2^4*x3^21+60081748212*x1^2*x2^3*x3^22+6723066792*x1^2*x2^2* x3^23+441714948*x1^2*x2*x3^24+11844090*x1^2*x3^25+438670*x1*x2^26+23688180*x1* x2^25*x3+441714948*x1*x2^24*x3^2+4608563571*x1*x2^23*x3^3+32044325724*x1*x2^22* x3^4+162262569876*x1*x2^21*x3^5+630878128824*x1*x2^20*x3^6+1949875912176*x1*x2^ 19*x3^7+4907893594934*x1*x2^18*x3^8+10236913446636*x1*x2^17*x3^9+17920117063620 *x1*x2^16*x3^10+26568767196693*x1*x2^15*x3^11+33569754912480*x1*x2^14*x3^12+ 36277357604136*x1*x2^13*x3^13+33569754912480*x1*x2^12*x3^14+26568767196693*x1* x2^11*x3^15+17920117063620*x1*x2^10*x3^16+10236913446636*x1*x2^9*x3^17+ 4907893594934*x1*x2^8*x3^18+1949875912176*x1*x2^7*x3^19+630878128824*x1*x2^6*x3 ^20+162262569876*x1*x2^5*x3^21+32044325724*x1*x2^4*x3^22+4608563571*x1*x2^3*x3^ 23+441714948*x1*x2^2*x3^24+23688180*x1*x2*x3^25+438670*x1*x3^26+438670*x2^26*x3 +11844090*x2^25*x3^2+152529987*x2^24*x3^3+1247030175*x2^23*x3^4+7267602840*x2^ 22*x3^5+32138190720*x2^21*x3^6+112073610312*x2^20*x3^7+316211239032*x2^19*x3^8+ 734829725830*x2^18*x3^9+1424369070610*x2^17*x3^10+2323642555077*x2^16*x3^11+ 3209874703017*x2^15*x3^12+3769169494200*x2^14*x3^13+3769169494200*x2^13*x3^14+ 3209874703017*x2^12*x3^15+2323642555077*x2^11*x3^16+1424369070610*x2^10*x3^17+ 734829725830*x2^9*x3^18+316211239032*x2^8*x3^19+112073610312*x2^7*x3^20+ 32138190720*x2^6*x3^21+7267602840*x2^5*x3^22+1247030175*x2^4*x3^23+152529987*x2 ^3*x3^24+11844090*x2^2*x3^25+438670*x2*x3^26)/x1^4/x2^4/x3^4/(x1^2*x2+x1^2*x3+ x1*x2^2-3*x1*x2*x3+x1*x3^2+x2^2*x3+x2*x3^2)^5 Finally, let's look what happens when all three players start out with the \ same capital, A, dollars each The expected duration is 2 A and in Maple notation A^2 The variance is 3 A (3 A - 3 A) -------------- 6 and in Maple notation 1/6*A*(3*A^3-3*A) The skewness is 6 4 2 1/2 1/2 1/2 (234 A - 270 A + 36 A ) 3 2 9 ---------------------------------------- 3 3/2 180 (A (3 A - 3 A)) and in Maple notation 1/180*(234*A^6-270*A^4+36*A^2)*3^(1/2)*2^(1/2)*9^(1/2)/(A*(3*A^3-3*A))^(3/2) Note that, as, A, goes to infinity it tends to 13/30*2^(1/2)*9^(1/2) `that, in decimals is`, 1.838477630 The kurtosis is 9 7 5 3 15633 A - 26082 A + 10017 A + 432 A --------------------------------------- 3 3 2 210 A (3 A - 3 A) and in Maple notation 1/210/A^3*(15633*A^9-26082*A^7+10017*A^5+432*A^3)/(3*A^3-3*A)^2 Note that, as, A, goes to infinity it tends to 579/70 `that, in decimals is`, 8.271428571 The scaled , 5, -th moment about the mean is 12 10 8 6 4 1/2 1/2 1/2 / (620946 A - 1232820 A + 665658 A - 55080 A + 1296 A ) 3 2 9 / / 2 3 5/2 (7560 A (A (3 A - 3 A)) ) and in Maple notation 1/7560/A^2*(620946*A^12-1232820*A^10+665658*A^8-55080*A^6+1296*A^4)*3^(1/2)*2^( 1/2)*9^(1/2)/(A*(3*A^3-3*A))^(5/2) Note that, as, A, goes to infinity it tends to 3833/420*2^(1/2)*9^(1/2) `that, in decimals is`, 38.71914702 The scaled , 6, -th moment about the mean is 15 13 11 9 7 (169249986 A - 400228290 A + 293083758 A - 59100030 A - 2694384 A 5 / 6 3 3 - 311040 A ) / (27720 A (3 A - 3 A) ) / and in Maple notation 1/27720/A^6*(169249986*A^15-400228290*A^13+293083758*A^11-59100030*A^9-2694384* A^7-311040*A^5)/(3*A^3-3*A)^3 Note that, as, A, goes to infinity it tends to 348251/1540 `that, in decimals is`, 226.1370130 The scaled , 7, -th moment about the mean is 18 16 14 12 (315669021210 A - 860901768090 A + 785750295330 A - 251257161870 A 10 8 6 1/2 1/2 1/2 / + 11252421180 A - 249434640 A - 263373120 A ) 3 2 9 / ( / 4 3 7/2 32432400 A (A (3 A - 3 A)) ) and in Maple notation 1/32432400/A^4*(315669021210*A^18-860901768090*A^16+785750295330*A^14-\ 251257161870*A^12+11252421180*A^10-249434640*A^8-263373120*A^6)*3^(1/2)*2^(1/2) *9^(1/2)/(A*(3*A^3-3*A))^(7/2) Note that, as, A, goes to infinity it tends to 14433883/40040*2^(1/2)*9^(1/2) `that, in decimals is`, 1529.415067 The scaled , 8, -th moment about the mean is 21 19 17 (5182222998267 A - 16030413661500 A + 17568839045394 A 15 13 11 9 - 7731551727900 A + 976186942731 A + 28212649920 A + 6523348608 A 7 / 9 3 4 - 19595520 A ) / (5405400 A (3 A - 3 A) ) / and in Maple notation 1/5405400/A^9*(5182222998267*A^21-16030413661500*A^19+17568839045394*A^17-\ 7731551727900*A^15+976186942731*A^13+28212649920*A^11+6523348608*A^9-19595520*A ^7)/(3*A^3-3*A)^4 Note that, as, A, goes to infinity it tends to 338508263/28600 `that, in decimals is`, 11835.95325 The scaled , 9, -th moment about the mean is 24 22 20 (32535217272268350 A - 112532663936428920 A + 143889245913464100 A 18 16 14 - 80646528096543240 A + 17155557717591150 A - 442729890635760 A 12 10 8 1/2 1/2 + 20080675332000 A + 18095334577920 A + 3725010374400 A ) 3 2 1/2 / 6 3 9/2 9 / (16540524000 A (A (3 A - 3 A)) ) / and in Maple notation 1/16540524000/A^6*(32535217272268350*A^24-112532663936428920*A^22+ 143889245913464100*A^20-80646528096543240*A^18+17155557717591150*A^16-\ 442729890635760*A^14+20080675332000*A^12+18095334577920*A^10+3725010374400*A^8) *3^(1/2)*2^(1/2)*9^(1/2)/(A*(3*A^3-3*A))^(9/2) Note that, as, A, goes to infinity it tends to 99177617047/4084080*2^(1/2)*9^(1/2) `that, in decimals is`, 103028.0977 The scaled , 10, -th moment about the mean is 27 25 (35513357382757366938 A - 135815515267227977790 A 23 21 + 198483600513866738868 A - 135226127326096589580 A 19 17 + 40635840583231591698 A - 3533267625435254070 A 15 13 11 - 49715623300688352 A - 9023638257365760 A - 257413072234752 A 9 / 12 3 5 + 1108413534412800 A ) / (146659312800 A (3 A - 3 A) ) / and in Maple notation 1/146659312800/A^12*(35513357382757366938*A^27-135815515267227977790*A^25+ 198483600513866738868*A^23-135226127326096589580*A^21+40635840583231591698*A^19 -3533267625435254070*A^17-49715623300688352*A^15-9023638257365760*A^13-\ 257413072234752*A^11+1108413534412800*A^9)/(3*A^3-3*A)^5 Note that, as, A, goes to infinity it tends to 128876105496249/129329200 `that, in decimals is`, 996496.5800 This ends this paper, that took , 46.317, to generate