Prove that
` P(n,n) = P(n,n-1)`

Prove that : P(n,n)= 2P (n,n -2)

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

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

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

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

A student was asked to prove a statement P(n) by using the principle of mathematical induction. He proved that P(n) Rightarrow P(n+1) for all n in N and also that P(4) is true: On the basis of the above he can conclude that P(n) is true.

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

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 5 P(n, 4)=6 P(n ,3) n?

If n be a positive integer and P_n denotes the product of the binomial coefficients in the expansion of (1 +x)^n , prove that (P_(n+1))/P_n=(n+1)^n/(n!) .