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

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

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

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

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)

Let P_(m) stand for mP_(m) then 1+P_(1)+2P_(2)+3P_(3)+.......+nP_(n) is equal to

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.