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_1xx^n C_2`

The smallest natural number n, such that the coefficient of x in the expansion of (x^(2) + (1)/(x^(3)))^(n) is .^(n)C_(23) , is

If |x|<1, then the coefficient of x^n in expansion of (1+x+x^2+x^3+)^2 is a. n b. n-1 c. n+2 d. n+1

If |x|<1, then the coefficient of x^n in expansion of (1+x+x^2+x^3+)^2 is a. n b. n-1 c. n+2 d. n+1

Find the coefficient of x^4 in the expansion of (1+x)^n (1-x)^n . Deduce that C_2=C_0C_4-C_1C_3+C_2C_2-C_3C_1+ C_4 C_0 .

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)

If |x|<1, then the coefficient of x^(n) in expansion of (1+x+x^(2)+x^(3)+...)^(2) is n b.n-1 c.n+2 d.n+1

If C_0,C_1,C_2..C_n denote the coefficients in the binomial expansion of (1 +x)^n , then C_0 + 2.C_1 +3.C_2+. (n+1) C_n

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) .

If C_0, C_1, C_2 ,……..C_n are the coefficient in the expansion of (1 + x)^n then show that C_0 C_r + C_1 C_(r + 1) + C_2 C_(r + 2) + ………..+ C_(n-r).C_n = ((2n)!)/((n-r)!(n+r)!)

If C_(0), C_(1), C_(2),..., C_(n) denote the binomial coefficients in the expansion of (1 + x)^(n) , then . 1^(2). C_(1) - 2^(2) . C_(2)+ 3^(2). C_(3) -4^(2)C_(4) + ...+ (-1).""^(n-2)n^(2)C_(n)= .