Q102 / 170 · Manipal (MET) 2021 · Mathematics

The coefficient of $x$ in the expansion of $\left(1+x+x^2+x^3\right)^{-3}$ is

  • A
    6
  • B
    9
  • C
    5
  • Correct answer
    D
    -3

Solution

$\left(1+x+x^2+x^3\right)^{-3}=\left[\frac{\left(x^4-1\right)}{x-1}\right]^{-3}$ $\begin{aligned} & =\left(x^4-1\right)^{-3}(x-1)^3 \\ & =(-1)^{-3}\left(1-x^4\right)^{-3}(x-1)^3 \\ & =(-1)\left(1+3 x^4+\ldots\right)\left(x^3-3 x^2+3 x-1\right)\end{aligned}$ $\therefore$ Coefficient of $x$ in $\left(1+x+x^2+x^3\right)^{-3}$ is -3 .

Hence, the correct answer is option: 'D'.

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