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

If (1 + x)^(n) = C_(0) = C_(1) x + C_(2) x^(2) + …+ C_(n) x^(n) , find the values of the following underset(0leile jlen)(sumsum)C_(i)C_(j)

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) , find the values of the following (sumsum)_(0leile jlen)(i +j)(C_(i)pmC_(j) )^(2)

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

If (1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + C_(3) x^(3) + …+ C_(n) x^(n) , find the sum of the series (C_(0))/(2) -(C_(1))/(6) + (C_(2))/(10) + (C_(3))/(14) -...+ (-1)^(n) (C_(n))/(4n+2) .

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 :

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 (x + a_(1)) (x + a_(2)) (x + a_(3)) …(x + a_(n)) = x^(n) + S_(1) x^(n-1) + S_(2) x^(n-2) + …+ S_(n) where , S_(1) = sum_(i=0)^(n) a_(i), S_(2) = (sumsum)_(1lei lt j le n) a_(i) a_(j) , S_(3) (sumsumsum)_(1le i ltk le n) a_(i) a_(j) a_(k) and so on . If (1 + x)^(n) = C_(0) + C_(1) x + C_(2)x^(2) + ...+ C_(n) x^(n) the cefficient of x^(n) in the expansion of (x + C_(0))(x + C_(1)) (x + C_(2))...(x + C_(n)) is

If (1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) +… + C_(n) x^(n) , prove that C_(0) + 2C_(1) + 3C_(2) + …+ (n+1)C_(n) = (n+2)2^(n-1) .

If (1+x)^n = C_0 + C_1x + C_2x^2 + ……..+C_n.x^n then find C_1 - C_3 + C_5 + ……