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

(1 - n)

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

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(n - 1)

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

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Solution

The correct option is C $(- 1)^{n} (1 - n)$
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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x_{1}, x_{2}, x_{3}, x_{4} . . . . x_{2 n + 1}

[\frac{(x_{2 n + 1} - x_{1})}{(x_{2 n + 1} + x_{1})}] + [\frac{(x_{2 n} - x_{2})}{(x_{2 n} + x_{2})}] . . . . . . . . . . . . . . . . . . . . [\frac{(x_{n + 2 - x_{n}})}{(x_{n + 2} + x_{n})}]

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