Find the coefficient of `x^n` in the polynomial `(x+^n C_0)(x+3^n C_1)xx(x+5^n C_2)[x+(2n+1)^n C_n]dot`

Find the coefficient of x in polynomial (x+2n+1C_(0))(x+^(2n+1)C_(1))......(x+^(2n+1)C_(n))

The coefficient of x^(n) in the polynomial (x+""^(2n+1)C_(0))(X+""^(2n+1)C_(1)) (x+""^(2n+1)C_(2))……(X+""^(2n+1)C_(n)) is

The coefficient of x^(n) in the polynomial (x+""^(2n+1)C_(0))(X+""^(2n+1)C_(1)) (x+""^(2n+1)C_(2))……(X+""^(2n+1)C_(n)) is

If C_(n) is the coefficient of x^(n) in the expansion of (1+x)^(n) then C_(1)+2. C_(2)+3. C_(3)+..+n. C_(n)

Find the coefficient of x^(n-r) in the expansion of (x+1)^n (1+x)^n . Deduce that C_0C_r+C_1C_(r-1)+......+C_(n-r) C_n= ((2n!))/((n+r)!(n-r)!) .

Show that the coefficient of x^(n)y^(n) in the expansion of {(1+x)(1+y)(x+y)}^(n) is C_(0)^(3)+C_(1)^(3) +C_(2)^(3) +…C_(n)^(3) .

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 xC_(0)-(x -1) C_(1)+(x-2)C_(2)-(x -3)C_(3)+...+(-1)^(n) (x -n) C_(n)=

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)=