prove that `1P_1+2.2P_2+3.3P_3+.........+n.nP_n=(n+1)P_(n+1)-1`

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

Let nP_(r) denote the number of permutations of n differenttaken r at a time.Then,prove that 1+1*1P_(1)+2*2P_(2)+3*3P_(3)+......+n*nP_(n)=n+1P_(n+)

If P_(m) stands for mP_(m), then prove that: 1+1.P_(1)+2.P_(2)+3.P_(3)+...+n.P_(n)=(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) .

Prove that: 1.P (1,1)+2.P(2,2)+3.P(3,3)+...+n.P(n,n)=P(n+1,n+1)-1

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

2.^(n)P_(3)=^(n+1)P_(3)

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 leq 3 .