The coefficient of $x^{n}$ in expansion of $(1 + x) {(1 - x)}^{n}$ , is

(n - 1)

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{(- 1)}^{n} (1 - n)

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{(- 1)}^{n - 1} {(n - 1)}^{2}

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{(- 1)}^{n - 1} n

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Solution

The correct option is C ${(- 1)}^{n} (1 - n)$
The coefficient of $x^{n}$ in expansion of $(1 + x) {(1 - x)}^{n}$
$=$ Coefficient of $x^{n} +$ Coefficient of $x^{n - 1}$
$= {(- 1)}^{n} \frac{n!}{n! 0!} + {(- 1)}^{n - 1} \frac{n!}{1! (n - 1)!}$
$= {(- 1)}^{n} [\frac{n!}{n! 0!} - \frac{n!}{1! (n - 1)!}]$
$= {(- 1)}^{n} [\frac{n!}{n! 0!} - \frac{n!}{1! (n - 1)!}]$
$= {(- 1)}^{n} (1 - n)$

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