Show that ` ""^(n)P_r =""^(n-1) P_r + r, ^(n-1) P_(r-1) ` Where the symbols have their usual meanings.

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

Let f(n)=|nn+1n+2^n P_n^(n+1)P_(n+1)^n P_n^n C_n^(n+1)C_(n+1)^(n+2)C_(n+2)| , where the symbols have their usual meanings. Then f(n) is divisible by n^2+n+1 b. (n+1)! c. n ! d. none of these

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

Prove that "^n C_r+^(n-1)C_r+...+^r C_r=^(n+1)C_(r+1) .

Prove that: (i) (.^(n)P_(r))/(.^(n)P_(r-2)) = (n-r+1) (n-r+2)

Prove that P(n,r) = (n- r+1) P(n,r-1)

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

If ^n P_r=^n P_(r+1)a n d^n C_r=^n C_(r-1,) then the value of n+r is.

If ""^(n)P_(r) = 840 and ""^(n)C_(r) = 35 , then r is equal to ………… .

If n P_r=^n P_(r+1) and n C_r=^n C_(r-1,) then the value of (n+r) is.