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

Prove that: P(1,1)+2.P(2,2)+3.P(3,3)++ndot P(n,n)=P(n+1,n+1)-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 1P_(1)+2.2P_(2)+3.3P_(3)+.......+n.nP_(n)=(n+1)P_(n+1)-1

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)!

If P_m stands for ^m P_m , then prove that: 1+1. P_1+2. P_2+3. P_3++ndotP_n=(n+1)!

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 .

If P(2n+1, n-1) : P(2n-1, n) = 3 : 5 then

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) .

It is tossed n xx.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) for all n<=3.

If P(n ,5): P(n ,3)=2:1 find n