Find the coefficient of $x^{n}$ in the expansion of $(1 + x) (1 - x)^{n}$ .

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Solution

Coefficient of $x^{n}$ in $(1 + x) (1 - x)^{n}$

$=$ Coefficient of $x^{n}$ in $(1 - x)^{n}$ $+$ Coefficient of $x^{n - 1}$ in $(1 - x)^{n}$

$= (- 1)^{n}$ $^{n} C_{n}$ + $(- 1)^{n - 1}$ $^{n} C_{n - 1}$

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

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