Question

The value of $1.C_1+3.C_3+5.C_5+7.C_7+....where{C}_0,C_1,C_2.....C_n$ are binomial coefficients in the expansion of $(1+x{)}^n$ is:

• A. n∙2^n-1
• B. n∙2^n-2
• C. n(n-1)2^n-1
• D. None of these

Answer 1

The correct option is B

n∙2^n-2

$C_0+C_{1}x+C_2{x}^2+...+C_n{x}^n=(1+x{)}^n$ .

Differentiating w.r.t. x,

$C_1+2C_{2}x+3C_3{x}^2+...+n{C}_n{x}^{n-1}=n(1+x{)}^{n-1}$

Putting x = 1 and x = -1 and then adding.

$2[C_1+3C_3+5C_5+....]=n{2}^{n-1}$