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Q.
The coefficient of $x$ in the expansion of $\left(1+x+x^{2}+x^{3}\right)^{-3}$ is
ManipalManipal 2017
Solution:
$\left(1+x+x^{2}+x^{3}\right)^{-3}=\left[\frac{\left(x^{4}-1\right)}{x-1}\right]^{-3}$
$=\left(x^{4}-1\right)^{-3}(x-1)^{3}$
$=(-1)^{-3}\left(1-x^{4}\right)^{-3}(x-1)^{3}$
$=(-1)\left(1+3 x^{4}+\ldots\right)\left(x^{3}-3 x^{2}+3 x-1\right)$
Coefficient of $x$ in $\left(1+x+x^{2}+x^{3}\right)^{-3}$ is $-3$