Question

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

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

Answer 1

Answer: (B)

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