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(n,n) = P(n,n-1)

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

Prove the following: P(n , n)=P(n , n-1)

If P(n-1,\ 3): P(n ,4)=1:9 find n

If P(n) is the statement n^2-n+41 is prime. Prove that P(1),\ P(2)a n d\ P(3) are true. Prove also that P(41) is not true.

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)!

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 .

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

Prove the following: P(n , r)=P(n-1, r)+rdotP(n-1,\ r-1)

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