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

Prove that if r<=s<=n, then ^(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))

Prove that ""^(n)P_(r )= ""^(n)C_(r )*^rP_(r ) .

Let ."""^(n)P_(r) denote the number of permutations of n different things taken r at a time . Then , prove that 1+1."""^1P_(1) + 2 ."""^(2)P_(2) + 3."""^(3)P_(3) +.....+ n . """^(n)P_(n) = . """^(n+1)P_(n+1)

Prove that .^(n-1)P_(r)+r.^(n-1)P_(r-1)=.^(n)P_(r)

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

Prove that ""^(n)C_(r ) ""^(r ) C_(s)= ""^(n)C_(s) ""^(n-s)C_(r-s)