Prove that if `rlt=slt=n ,t h e n^n P_s` is divisible by `^n P_r` .

Prove that quad if r<=s<=n then ^(^^)n_(P_(s)) is divisible by ^(^^)n_(P_(r))

If r lt s le n " then prove that " ^(n)P_(s) " is divisible by "^(n)P_(r).

Prove that (n!)! is divisible by (n!)^((n-1)!)

Prove that the Principle of Mathematical Induction does not apply to the following: (i) P(n) : n^(3)+ n is divisible by 3" (ii) P(n): n^(3) le 100

If p is a fixed positive integer,prove by induction that p^(n+1)+(p+1)^(2n-1) is divisible by P^(2)+p+1 for all n in N

If P(n) stands for the statement n(n+1)(n+2) is divisible by 3, then what is P(4).

Prove that P(n,n)=2.P(n,n-2)