Show that `1+^1P_1+2.^2P_2+3.^3P_3+....+n.^nP_n=^(n+1)P_(n+1)`.

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

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

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

A coin has probability p of showing head when tossed. It is tossed n times. Let P_n denote the probability that no two (or more) consecutive heads occur. Prove that P_1 = 1,P_2 = 1 - p^2 and P_n= (1 - p) P_(n-1) + p(1 - p) P_(n-2) for all n geq 3 .

Prove that .^(2n)P_(n)={1.3.5.....(2n-1)}.2n

For n being a natural number prove that 1.1!+2.2!+3.3!+.....+n.n! =(n+1)!-1 by applying P.M.I

If 1*1!+2*2!+3*3!+ . . .+n*n ! =(n+1)!-1 then show that, 1*1!+2*2!+3*3!+ . . . +n*n! +(n+1)(n+1)! =(n+2)!-1

If a_1,a_2,a3,...,a_n are in A.P then show that 1/(a_1a_2)+1/(a_2a_3)+1/(a_3a_4)+...+1/(a_(n-1)a_n)=(n-1)/(a_1a_n)

A is a set containing n elements. A subset P_1 is chosen and A is reconstructed by replacing the elements of P_1 . The same process is repeated for subsets P_1,P_2,....,P_m with m>1 . The number of ways of choosing P_1,P_2,....,P_m so that P_1 cup P_2 cup....cup P_m=A is (a) (2^m-1)^(mn) (b) (2^n-1)^m (c) (m+n)C_m (d) none of these

Applying the principle of mathematical induction (P.M.I.) prove that 1^2+2^2+3^2+.......+n^2=(n(n+1)(2n+1))/6