For all positive integers `n ,` show that `\ ^(2n)C_n+\ ^(2n)C_(n-1)=1/2(\ ^(2n+2)C_(n+1))` .

Prove that: \ ^(2n)C_n=(2^n[1. 3. 5 (2n-1)])/(n !)

If n and k are positive integers, show that 2^k( .^n C_0)(.^n C_k)-2^(k-1)(.^n C_1)(.^(n-1) C_k-1)+2^(k-2)(.^n C_2)((n-2k-2))_dot-...+ (-1)^k(^n C_k)+(.^(n-k) C_0)=(.^n C_k)w h e r e(.^n C_k) stands for .^n C_k.

If m,n,r are positive integers such that r lt m,n, then ""^(m)C_(r)+""^(m)C_(r-1)""^(n)C_(1)+""^(m)C_(r-2)""^(n)C_(2)+...+ ""^(m)C_(1)""^(n)C_(r-1)+""^(n)C_(r) equals

If n is a positive integer then ^nC_r+^nC_(r+1)=^(n+1)^C_(r+1) Also coefficient of x^r in the expansion of (1+x)^n=^nC_r. In an identity in x, coefficient of similar powers of x on the two sides re equal. On the basis of above information answer the following question: If n is a positive integer then ^nC_n+^(n+1)C_n+^(n+2)C_n+.....+^(n+k)C_n= (A) ^(n+k+1)C_(n+2) (B) ^(n+k+1)C_(n+1) (C) ^(n+k+1)C_k (D) ^(n+k+1)C_(n-2)

If n is a positive integer then ^nC_r+^nC_(r+1)=^(n+1)^C_(r+1) Also coefficient of x^r in the expansion of (1+x)^n=^nC_r. answer the following question: If n is a positive integer then ^nC_n+^(n+1)C_n+^(n+2)C_n+.....+^(n+k)C_n= (A) ^(n+k+1)C_(n+2) (B) ^(n+k+1)C_(n+1) (C) ^(n+k+1)C_k (D) ^(n+k+1)C_(n-2)

Let n be a positive integer such that sin (pi/(2n))+cos (pi/(2n))= sqrt(n)/2 then (A) n=6 (B) n=1,2,3,….8 (C) n=5 (D) none of these

If n is a positive integer, then show tha 3^(2n+1)+2^(n+2) is divisible by 7 .

For a fixed positive integer n , if =|n !(n+1)!(n+2)!(n+1)!(n+2)!(n+3)!(n+2)!(n+3)!(n+4)!| , then show that [/((n !)^3)-4] is divisible by ndot

If n is a positive integer, prove that underset(r=1)overset(n)sumr^(3)((""^(n)C_(r))/(""^(n)C_(r-1)))^(2)=((n)(n+1)^(2)(n+2))/(12)

Prove that sum_(r=0)^(2n) r.(""^(2n)C_(r))^(2)= 2.""^(4n-1)C_(2n-1) .