`sum sum sum _(0 le i lt j lt k le n) 1` is equal to

If (1 + x)^(n) = C_(0) + C_(1)x + C_(2)x^(2) +.................+ C_(n)x^(n) then show that the sum of the products of the C_(i)'s taken two at a time represents by : {:(" "sum" "sum" " c_(i)c_(j)),(0 le i lt j le n ):} is equal to 2^(2n-1)-(2n!)/(2.n!.n!)

sum sum_ (0 <= i

The value of (sum sum sum sum sum)_(0<=i

If (1+x)^n=C_(0)C_1c+C_(2)x^2+…..+C_(n)x^n then show that the sum of the products of the C_(i) taken two at a time represented by : Sigma_(0 le I lt) Sigma_( j le n) C_(i)C_(j) "is equal to " 2^(2n-1)-(2n!)/(2.n! n !)

The value of the expansion (sumsum)_(0 le i lt j le n) (-1)^(i+j-1)"^(n)C_(i)*^(n)C_(j)=

The value of the expansion (sumsum)_(0 le i lt j le n) (-1)^(i+j-1)"^(n)C_(i)*^(n)C_(j)=

The value of the expansion (sumsum)_(0 le i lt j le n) (-1)^(i+j-1)"^(n)C_(i)*^(n)C_(j)=

Statement 1: sum sum_(0le ilt j le n)(i/ (^n c_i)+j/(^nc_j)) is equal to(n^2)/2a , where a ,sum_(r="0)^(n) 1/(^n"" c_r)="" .="" statement 2:sum_(r=0)^(n) r/(^n" c_r)="sum_(r=0)^(n)(n-r)/(^n" .

If n ge3 and 1,alpha_1, alpha_2, alpha_3....alpha_(n-1) are the , nth roots of unity then find the value of sum_(1 le i lt j le (n-1)) (alpha_i alpha_j )