Question

Let $(1+x)(1+x+x^2)(1+x+x^2+x^3)......(1+x+x^2+....+x^{30})=a_0+a_{1}x+a_2{x}^2+........+a_{465}{x}^{465}$

then sum of $a_0+a_2+a_4+........+a_{464}$ is

• A. $\left(31\right)!$
• B. $\frac{\left(31\right)!}{2}$
• C. $\left(30\right)!$
• D. $\frac{\left(60\right)!}{2}$

Answer 1

The correct option is B $\frac{\left(31\right)!}{2}$

Putting $x=1$
$a_0+a_1+a_2+.......+a_{464}=\left(31\right)!.....\left(i\right)$
Putting $x=-1$
$a_0-a_1+a_2-.......+a_{464}=\mathrm{0.....}\left(ii\right)$
Adding both the equation,
$a_0+a_2+a_4+......=\frac{\left(31\right)!}{2}$