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

Question

Aicharya · Accepted Answer

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

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

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