Find the sum `1C_0+2C_1+3C_2++(n+1)C_n ,w h e r eC_r=^n C_rdot`

Find the sum 1xx2xx C_(1)+2xx3C_(2)+..+n(n+1)C_(n), where C_(r)=^(n)C_(r)

If (1+x)^n=C_0+C_1x+C_2x^2+…+C_nx^n show that C_1-2C_2+3C_3-4C_4+…+(-1)^(n-1) n.C_n=0 where C_r=^nC_r .

If (1+x)^n=C_0+C_1x+C2x2++C_n x^n ,t h e nC_0-(C_0+C_1+)+(C_0+C_1+C_2)-(C_0+C_1+C_2+C_3)+(-1)^(n-1)(C_0+C_1+ C_(n-1)),w h e r en is even integer is a positive value a negative value divisible by2^(n-1) divisible by2^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_(r) = ""^(n)C_(r) and (C_(0) + C_(1)) (C_(1) + C_(2)) … (C_(n-1) + C_(n)) = k ((n +1)^(n))/(n!) , then the value of k, is

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) =