Question

The coefficient of $x^{4}$ in the expansion of $\frac{1}{(1-x)(1-2 x)(1-3 x)}$ is

• A. 602
• B. 301
• C. $\frac{601}{2}$
• D. 302

Answer 1

Answer: (B)

Since, $\frac{1}{(1-x)(1-2 x)(1-3 x)}$
$=(1-x)^{-1}(1-2 x)^{-1}(1-3 x)^{-1}$
$=\left(1+x+x^{2}+x^{3}+x^{4}\right)\left(1+2 x+4 x^{2}+8 x^{3}+16 x^{4}\right)$
$\left(1+3 x+9 x^{2}+27 x^{3}+81 x^{4}\right)$
[Expand till $x^{4}$ -term because coefficient of $x^{4}$ is required and $(1-a x)^{-n}$
$\left.=1+a x+a^{2} x^{2}+a^{3} x^{3}+\ldots \ldots \ldots . .\right]$
So, coefficient of $x^{4}$ is
$81+54+36+24+16+27+18+12+8+9+6+4+3+2+1=301$