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`.

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)!) .

If C_(0) , C_(1), C_(2), …, C_(n) are the binomial coefficients in the expansion of (1 + x)^(n) , prove that (C_(0) + 2C_(1) + C_(2) )(C_(1) + 2C_(2) + C_(3))…(C_(n-1) + 2C_(n) + C_(n+1)) ((n-2)^(n))/((n+1)!) prod _(r=1)^(n) (C_(r-1) + C_(r)) .

If C_0,C_1,C_2,........, C_n denote the coefficients in the expansion of (1+ x)^n , then the value of : C_1+2C_2+3 C_3+..... +nC_n is :

If C_1,C_2,C_3,C_4 are the coefficients of any four consecutive terms in the expansion of (1+ x)^n , prove that : C_1/(C_1+C_2)+C_3/(C_3+C_4)=(2C_2)/(C_2+C_3) .

If C_(0), C_(1), C_(2),…, C_(n) denote the binomial coefficients in the expansion of (1 + x)^(n) , then sum_(r=0)^(n)(r+1)(C_(r))

Prove that : ^2C_1+ ^3C_1+ ^4C_1= "^3C_2+ ^4C_2 .

Find the order of the family of curves y=(c_1+c_2)e^x+c_3e^(x+c_4)

(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) .

The coefficient of x^r [0lt=rlt=(n-1)] in the expansion of (x+3)^(n-1)+(x+3)^(n-2)(x+2)+(x+3)^(n-3)(x+2)^2+.... +(x+2)^(n-1) is a.^n C_r(3^r-2^n) b.^n C_r(3^(n-r)-2^(n-r)) c. ^n C_r(3^r+2^(n-r)) d. none of these

If a,b,c,d be four consecutive coefficients in the binomial expansion of (1+x)^(n) , then value of the expression (((b)/(b+c))^(2)-(ac)/((a+b)(c+d))) (where x gt 0 and n in N ) is