Prove that if `r le s le n`, then `""^nP_s` is divisible by `""^nP_r` .

If P(n) is the statement that the sum of first n natural numbers is divisible by n +1, prove that P(2r) is true for all r = 1, 2, 3,...... .

If P(n) is the statement “sum of first n natural numbers is divisible by n + 1", prove that P(r + 1) is true if P(r) is true.

If n=8 and r= 3, then the value of ""^nP_r is :

Prove that "^nP_r= ^nC_r ^rP_r .

If ""^nC_r= ""^nC_s , then n is:

Prove that the relation R defined in the set A= { x: x in Z, 0 le x le 12} as R = {(a, b) : abs(a-b) . is divisible by 5} is an equivalence relation.

Prove that the relation R defined in the set A= { x: x in Z, 0 le x le 12} as R = {(a, b) : abs(a-b) . is divisible by 3} is an equivalence relation.

Prove that : "^nP_n= 2^nP_(n-2) .

If ""^nP_4=5 ""^nP_3 , then n is :