It is ok to post!
# Name:Treasa Bency Biju Jose
# Date: 09-22-2020
# Assignment #5
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I.
# To Prove: p(n) = n* p(n-1)
# p(n) = |P(n)|

with(combinat):
P := n -> permute(n);
	P := n -> permute(n)



P(2);
                        [[1, 2], [2, 1]]

p := n -> nops(P(n));
   p := n -> nops(P(n)) 

p(3);
                               6

p(2);
                               2


# p(3) = 3 * p(2)
# p(3) = 3 * 2 = 6
# Hence proved


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IV.
If some one is clever and good-lookingbit neither strong nor kind, make sure that he or she are counted exactly once, by finding out
How many time he is INCLUDED in line 1: 4 
How many time he is EXCLUDED in line 2: 0
How many time he is INCLUDED (again) in line 3: 4
How many time he is EXCLUDED (again) in line 4: 0

PIE4 := proc(U, C, S, L, K) local m; 
if not type(U, set) then 
	RETURN(FAIL); end if; 
if not type(C, set) then 
	RETURN(FAIL); end if; 
if not type(S, set) then 
	RETURN(FAIL); end if; 
if not type(L, set) then 
	RETURN(FAIL); end if; 
if not type(K, set) then 
	RETURN(FAIL); end if; 
if (C intersect S intersect L intersect K) = {} then 
	RETURN(FAIL); end if; 
m := ((nops(C) + nops(S) + nops(L) + nops(K)) + 
     -(nops(C intersect S) + nops(C intersect L) + nops(C intersect K) + nops(S intersect L) + nops(S intersect K) + nops(L intersect K)) 
     +(nops(C intersect S intersect L) + nops(S intersect L intersect K) + nops(C intersect L intersect K) + nops(C intersect S intersect K))
     - nops(C union S union L union K):
end proc:
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