#Please do not post homework
#Ravali Bommanaboina, 10/18/20, Assignment 11
#Question 1
# The number of ways to assemble 11 Russian dolls of different sizes into 5 different towers is the number of permutations of [1,...,11] with exactly 5 cycles.
# the maple code for cnk(n,k) can be used however maple cannot compute this fast enough.
# although cnk(11,5) would give the correct answer because each "tower" would correlate to one cycle
#Question 2
xn:=proc(x,n) local i,result:
result:=x;
for i from 1 to n-1 do
	result:=result*(x-i)
end do:
return(result):
end:


Axn:=proc(x,n) local i,result:
result:=0:
for i from 0 to n do
result:=result + Snk(n,i)*xn(x,i)
end do:
return(result):
end:

#seq(Axn(x, i), i = 1 .. 20); -> x, x + x*(x - 1), x + 3*x*(x - 1) + x*(x - 1)*(x - 2), x + 7*x*(x - 1) + 6*x*(x - 1)*(x - 2) + x*(x - 1)*(x - 2)*(x - 3), x + 15*x*(x - 1) + 25*x*(x - 1)*(x - 2) + 10*x*(x - 1)*(x - 2)*(x - 3) + x*(x - 1)*(x - 2)*(x - 3)*(x - 4), x + 31*x*(x - 1) + 90*x*(x - 1)*(x - 2) + 65*x*(x - 1)*(x - 2)*(x - 3) + 15*x*(x - 1)*(x - 2)*(x - 3)*(x - 4) + x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5), x + 63*x*(x - 1) + 301*x*(x - 1)*(x - 2) + 350*x*(x - 1)*(x - 2)*(x - 3) + 140*x*(x - 1)*(x - 2)*(x - 3)*(x - 4) + 21*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5) + x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6), x + 127*x*(x - 1) + 966*x*(x - 1)*(x - 2) + 1701*x*(x - 1)*(x - 2)*(x - 3) + 1050*x*(x - 1)*(x - 2)*(x - 3)*(x - 4) + 266*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5) + 28*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6) + x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7), x + 255*x*(x - 1) + 3025*x*(x - 1)*(x - 2) + 7770*x*(x - 1)*(x - 2)*(x - 3) + 6951*x*(x - 1)*(x - 2)*(x - 3)*(x - 4) + 2646*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5) + 462*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6) + 36*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7) + x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8), x + 511*x*(x - 1) + 9330*x*(x - 1)*(x - 2) + 34105*x*(x - 1)*(x - 2)*(x - 3) + 42525*x*(x - 1)*(x - 2)*(x - 3)*(x - 4) + 22827*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5) + 5880*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6) + 750*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7) + 45*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8) + x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9), x + 1023*x*(x - 1) + 28501*x*(x - 1)*(x - 2) + 145750*x*(x - 1)*(x - 2)*(x - 3) + 246730*x*(x - 1)*(x - 2)*(x - 3)*(x - 4) + 179487*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5) + 63987*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6) + 11880*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7) + 1155*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8) + 55*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9) + x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9)*(x - 10), x + 2047*x*(x - 1) + 86526*x*(x - 1)*(x - 2) + 611501*x*(x - 1)*(x - 2)*(x - 3) + 1379400*x*(x - 1)*(x - 2)*(x - 3)*(x - 4) + 1323652*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5) + 627396*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6) + 159027*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7) + 22275*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8) + 1705*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9) + 66*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9)*(x - 10) + x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9)*(x - 10)*(x - 11), x + 4095*x*(x - 1) + 261625*x*(x - 1)*(x - 2) + 2532530*x*(x - 1)*(x - 2)*(x - 3) + 7508501*x*(x - 1)*(x - 2)*(x - 3)*(x - 4) + 9321312*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5) + 5715424*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6) + 1899612*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7) + 359502*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8) + 39325*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9) + 2431*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9)*(x - 10) + 78*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9)*(x - 10)*(x - 11) + x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9)*(x - 10)*(x - 11)*(x - 12), x + 8191*x*(x - 1) + 788970*x*(x - 1)*(x - 2) + 10391745*x*(x - 1)*(x - 2)*(x - 3) + 40075035*x*(x - 1)*(x - 2)*(x - 3)*(x - 4) + 63436373*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5) + 49329280*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6) + 20912320*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7) + 5135130*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8) + 752752*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9) + 66066*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9)*(x - 10) + 3367*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9)*(x - 10)*(x - 11) + 91*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9)*(x - 10)*(x - 11)*(x - 12) + x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9)*(x - 10)*(x - 11)*(x - 12)*(x - 13), x + 16383*x*(x - 1) + 2375101*x*(x - 1)*(x - 2) + 42355950*x*(x - 1)*(x - 2)*(x - 3) + 210766920*x*(x - 1)*(x - 2)*(x - 3)*(x - 4) + 420693273*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5) + 408741333*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6) + 216627840*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7) + 67128490*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8) + 12662650*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9) + 1479478*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9)*(x - 10) + 106470*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9)*(x - 10)*(x - 11) + 4550*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9)*(x - 10)*(x - 11)*(x - 12) + 105*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9)*(x - 10)*(x - 11)*(x - 12)*(x - 13) + x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9)*(x - 10)*(x - 11)*(x - 12)*(x - 13)*(x - 14), x + 32767*x*(x - 1) + 7141686*x*(x - 1)*(x - 2) + 171798901*x*(x - 1)*(x - 2)*(x - 3) + 1096190550*x*(x - 1)*(x - 2)*(x - 3)*(x - 4) + 2734926558*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5) + 3281882604*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6) + 2141764053*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7) + 820784250*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8) + 193754990*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9) + 28936908*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9)*(x - 10) + 2757118*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9)*(x - 10)*(x - 11) + 165620*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9)*(x - 10)*(x - 11)*(x - 12) + 6020*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9)*(x - 10)*(x - 11)*(x - 12)*(x - 13) + 120*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9)*(x - 10)*(x - 11)*(x - 12)*(x - 13)*(x - 14) + x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9)*(x - 10)*(x - 11)*(x - 12)*(x - 13)*(x - 14)*(x - 15), x + 65535*x*(x - 1) + 21457825*x*(x - 1)*(x - 2) + 694337290*x*(x - 1)*(x - 2)*(x - 3) + 5652751651*x*(x - 1)*(x - 2)*(x - 3)*(x - 4) + 17505749898*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5) + 25708104786*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6) + 20415995028*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7) + 9528822303*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8) + 2758334150*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9) + 512060978*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9)*(x - 10) + 62022324*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9)*(x - 10)*(x - 11) + 4910178*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9)*(x - 10)*(x - 11)*(x - 12) + 249900*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9)*(x - 10)*(x - 11)*(x - 12)*(x - 13) + 7820*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9)*(x - 10)*(x - 11)*(x - 12)*(x - 13)*(x - 14) + 136*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9)*(x - 10)*(x - 11)*(x - 12)*(x - 13)*(x - 14)*(x - 15) + x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9)*(x - 10)*(x - 11)*(x - 12)*(x - 13)*(x - 14)*(x - 15)*(x - 16), x + 131071*x*(x - 1) + 64439010*x*(x - 1)*(x - 2) + 2798806985*x*(x - 1)*(x - 2)*(x - 3) + 28958095545*x*(x - 1)*(x - 2)*(x - 3)*(x - 4) + 110687251039*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5) + 197462483400*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6) + 189036065010*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7) + 106175395755*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8) + 37112163803*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9) + 8391004908*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9)*(x - 10) + 1256328866*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9)*(x - 10)*(x - 11) + 125854638*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9)*(x - 10)*(x - 11)*(x - 12) + 8408778*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9)*(x - 10)*(x - 11)*(x - 12)*(x - 13) + 367200*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9)*(x - 10)*(x - 11)*(x - 12)*(x - 13)*(x - 14) + 9996*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9)*(x - 10)*(x - 11)*(x - 12)*(x - 13)*(x - 14)*(x - 15) + 153*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9)*(x - 10)*(x - 11)*(x - 12)*(x - 13)*(x - 14)*(x - 15)*(x - 16) + x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9)*(x - 10)*(x - 11)*(x - 12)*(x - 13)*(x - 14)*(x - 15)*(x - 16)*(x - 17), x + 262143*x*(x - 1) + 193448101*x*(x - 1)*(x - 2) + 11259666950*x*(x - 1)*(x - 2)*(x - 3) + 147589284710*x*(x - 1)*(x - 2)*(x - 3)*(x - 4) + 693081601779*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5) + 1492924634839*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6) + 1709751003480*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7) + 1144614626805*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8) + 477297033785*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9) + 129413217791*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9)*(x - 10) + 23466951300*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9)*(x - 10)*(x - 11) + 2892439160*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9)*(x - 10)*(x - 11)*(x - 12) + 243577530*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9)*(x - 10)*(x - 11)*(x - 12)*(x - 13) + 13916778*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9)*(x - 10)*(x - 11)*(x - 12)*(x - 13)*(x - 14) + 527136*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9)*(x - 10)*(x - 11)*(x - 12)*(x - 13)*(x - 14)*(x - 15) + 12597*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9)*(x - 10)*(x - 11)*(x - 12)*(x - 13)*(x - 14)*(x - 15)*(x - 16) + 171*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9)*(x - 10)*(x - 11)*(x - 12)*(x - 13)*(x - 14)*(x - 15)*(x - 16)*(x - 17) + x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9)*(x - 10)*(x - 11)*(x - 12)*(x - 13)*(x - 14)*(x - 15)*(x - 16)*(x - 17)*(x - 18), x + 524287*x*(x - 1) + 580606446*x*(x - 1)*(x - 2) + 45232115901*x*(x - 1)*(x - 2)*(x - 3) + 749206090500*x*(x - 1)*(x - 2)*(x - 3)*(x - 4) + 4306078895384*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5) + 11143554045652*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6) + 15170932662679*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7) + 12011282644725*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8) + 5917584964655*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9) + 1900842429486*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9)*(x - 10) + 411016633391*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9)*(x - 10)*(x - 11) + 61068660380*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9)*(x - 10)*(x - 11)*(x - 12) + 6302524580*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9)*(x - 10)*(x - 11)*(x - 12)*(x - 13) + 452329200*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9)*(x - 10)*(x - 11)*(x - 12)*(x - 13)*(x - 14) + 22350954*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9)*(x - 10)*(x - 11)*(x - 12)*(x - 13)*(x - 14)*(x - 15) + 741285*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9)*(x - 10)*(x - 11)*(x - 12)*(x - 13)*(x - 14)*(x - 15)*(x - 16) + 15675*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9)*(x - 10)*(x - 11)*(x - 12)*(x - 13)*(x - 14)*(x - 15)*(x - 16)*(x - 17) + 190*x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9)*(x - 10)*(x - 11)*(x - 12)*(x - 13)*(x - 14)*(x - 15)*(x - 16)*(x - 17)*(x - 18) + x*(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9)*(x - 10)*(x - 11)*(x - 12)*(x - 13)*(x - 14)*(x - 15)*(x - 16)*(x - 17)*(x - 18)*(x - 19)

#Question 3
#We count the number of permutations of n with k cycles where n is in cycle of length m.
#We can let n be the first elements in our m-cycle. Then the second elements in our m-cycle will be one of the n−1 other options. Similarly, the third elements will have n−2 options (every thing but n and second elements we have picked) and so on. The number of ways to choose our m-cycle will be (n − 1)m−1. And since our permutation has k cycles, the remaining elements will be in a permutation with k − 1 cycles. This
#implies we have c(n − m, k − 1) choices for rest of elements.
#Therefore, number of permutations where n is in a m-cycle will be (n−1)m−1c(n−m,k−1). By summing
#up all possible m, we get our desired equality.

#Question 4
#GrabCycle(P,i): inputs a permutation P of {1, ...,n} and outputs the cycle belonging to i.
#We use the convention that it starts with the largest entry
#Try
#GrabCycle([3,1,4,2],2);
GrabCycle:=proc(P,i) local C,j:

#We view the cycle belnging to i as a list, starting at i
C:=[i]:

#We find where i points to, call it j
j:=P[i]:

while j<>i do
#sooner or later we will wind back at i (since the permutation is finite) so we append the new
#arrivals to the list C and rename j
C:=[op(C),j]:
j:=P[j]:
od:

#We look at the place where it it is max
j:=max[index](C):

#we rewrite the cycle with the largest entry first

[op(j..nops(C),C),op(1..j-1,C)]:

end:


#PtoC(P): Inputs a permutation P outputs its list of cycles. Try:
#PtoC([2,1,3,4]);
PtoC:=proc(P) local n,StillToDo,L,i,C,L1,T:

n:=nops(P):

StillToDo:={seq(i,i=1..n)}:

#L is a dynamically construcete list of cycles in the permutatib P, we start with the empty list
#until no one is left
L:=[]:

while StillToDo<>{} do
#we pick the smallest survivor
 i:=min(op(StillToDo)):
 #we grab its cycle
 C:=GrabCycle(P,i):

# We append C to L
 L:=[op(L),C]:

#We update StillToDo by kicking out the members of C

StillToDo:= StillToDo minus convert(C,set):
od:

#So far L is a list of cycles but arranged in a random order
#Now we arrange them according to the convention that the first (largest) entries are increasing
#L1 is the list of first (largest) entries of each cycle:
L1:=[seq(L[i][1],i=1..nops(L))]:

#We now make a table where T[a] is the cycle that starts with  a
for i from 1 to nops(L) do
 T[L[i][1]]:=L[i]:
od:

#Now we sort L1
L1:=sort(L1):

#Now we rewrite L with the above convention
[seq(T[L1[i]],i=1..nops(L1))]:

end:

cnk:=proc(n,k) local i,s,result:
s:=permute(n):
result:=0:
for i in s do
if(nops(PtoC(i))=k) then
result:=result+1:
end if:
end do:
return(result):
end: