Let `C_(r)` denote the binomial coefficient of `x^r` in the expansion of `(1+x)^(10).` If `1times2^(1)C_(1)+2times2^(2)C_(2)+3times2^(3)C_(3)+...+10times2^(10)C_(10)=alphatimes3^(9)` ,then `alpha` is equal to

Let ""^(n)C_(r) denote the binomial coefficient of x^(r) in the expansion of (1+x)^(n) . If underset(k=0)overset(10)sum (2^(2)+3k) ""^(n)C_(k)=alpha 3^(10)+beta. 2^(10), alpha, beta in R then alpha+beta is equal to..........

If C_(o),C_(1),C_(2)......,C_(n)C denote the binomial coefficients in the expansion of (1+x)^(n), then 1^(3)*C_(1)+2^(3)*C_(2)+3^(3)*C_(3)+...+n^(3)*C_(n),=

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

The coefficient of x^(4) in the expansion of (1+x+x^(2)+x^(3))^(n) is *^(n)C_(4)+^(n)C_(2)+^(n)C_(1)xx^(n)C_(2)

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)=

The coefficient of x^(20) in the expansion of (1+3x+3x^(2)+x^(3))^(20) is (1) .^(60)C_(40) (2) .^(30)C_(20) (3) .^(15)C_(2) (4) .^(60)C_(10)

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) =

The coefficient of x^(10) in the expansion of (1+x)^(2)(1+x^(2))^(3)(1+x^(3))^(4) is equal to

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)=