The coefficient of $x^{13}$ in the expansion of $(1 - x)^{5} (1 + x + x^{2} + x^{3})^{4}$ is

4

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- 4

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0

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none of these

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Solution

The correct option is A $4$
Given, $(1 - x)^{5} (1 + x + x^{2} + x^{3})^{4}$
$= (1 - x)^{5} ((1 + x) (1 + x^{2}))^{4}$
$= (1 - x) (1 - x^{2})^{4} (1 + x^{2})^{4}$
$= (1 - x) (1 - x^{4})^{4}$
$= (1 - x^{4})^{4} - x (1 - x^{4})^{4}$
Hence the coefficient of $x^{13}$ will be
$= - [(- 1)^{3}^{4} C_{3}]$
$= 4$

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