If `P_(4)` denote the produt of the binomial coefficients
in the expansion of `(1 + x)^(n)`, then `(P_(n+1))/(P_(n))` equals

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!) .

If C_(0), C_(1), C_(2), …, C_(n) denote the binomial coefficients in the expansion of (1 + x)^(n) , then sum_(0 ler )^(n)sum_(lt s len)^(n)C_(r)C_(s) =.

If C_(0), C_(1), C_(2), …, C_(n) denote the binomial coefficients in the expansion of (1 + x)^(n) , then sum_(r=0)^(n)sum_(s=0)^(n)C_(r)C_(s) =

If C_(0), C_(1), C_(2),…, C_(n) denote the binomial coefficients in the expansion of (1 + x)^(n) , then sum_(r=0)^(n)sum_(s=0)^(n)(C_(r) +C_(s))

If C_(0), C_(1), C_(2),..., C_(n) denote the binomial coefficients in the expansion of (1 + x)^(n) , then . 1^(2). C_(1) - 2^(2) . C_(2)+ 3^(2). C_(3) -4^(2)C_(4) + ...+ (-1).""^(n-2)n^(2)C_(n)= .

If 36, 84, 126 are three successive binomial coefficients in the expansion of (1+x)^(n) , find n.

If C_(0),C_(1), C_(2),...,C_(N) denote the binomial coefficients in the expansion of (1 + x)^(n) , then 1^(3). C_(1)-2^(3). C_(3) - 4^(3) . C_(4) + ...+ (-1)^(n-1)n^(3) C_(n)=

If C_(0), C_(1), C_(2),...,C_(n) denote the binomial coefficients in the expansion of (1 + x)^n) , then xC_(0)-(x -1) C_(1)+(x-2)C_(2)-(x -3)C_(3)+...+(-1)^(n) (x -n) C_(n)=

If ""(n)C_(0), ""(n)C_(1), ""(n)C_(2), ...., ""(n)C_(n), denote the binomial coefficients in the expansion of (1 + x)^(n) and p + q =1 sum_(r=0)^(n) r^(2 " "^n)C_(r) p^(r) q^(n-r) = .

If C_0, C_1,C_2 ..., C_n , denote the binomial coefficients in the expansion of (1 + x)^n , then C_1/2+C_3/4+C_5/6+...... is equal to