If `(1+x)^n=C_0+C_1x+C_2x^2+.......+C_n x^n` , then show that the sum of the products of the coefficients taken two at a time, represented by `sumsum_(0lt=iltjlt=n) ``"^nc_i``"^n c_j` is equal to `2^(2n-1)-((2n)!)/ (2(n !)^2)`

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!)

If (1+x)^n = C_0 + C_1x + C_2x^2 + ………. + C_n x^n , prove that : C_0 + 2C_1 + ….. + 2 ""^nC_n = 3^n

If (1 + x)^(n) = C_(0) = C_(1) x + C_(2) x^(2) + …+ C_(n) x^(n) , find the values of the following sumsum_(0 le i lt j le n)C_(i)

If (1 + x)^(n) = C_(0) = C_(1) x + C_(2) x^(2) + …+ C_(n) x^(n) , find the values of the following (sumsum)_(0leilt j le n)jC_(i)

If (1+x)^(n)=C_(0)+C_(1)x+C_(2)x^(2)+…..+C_(n)x^(n) , then the value of sumsum_(0lerltslen)(r*s)C_(r)C_(s) is :

If (1+x)^(n)=C_(0)+C_(1)x+C_(2)x^(2)+….+C_(n)x^(n) , then the value of sumsum_(0lerltslen)(C_(r)+C_(s))^(2) is :

(1 + x)^(n) = C_(0) + C_(1)x + C_(2) x^(2) + ...+ C_(n) x^(n) , show that sum_(r=0)^(n) C_(r)^(3) is equal to the coefficient of x^(n) y^(n) in the expansion of {(1+ x)(1 + y) (x + y)}^(n) .