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Binomial Expression

The coefficie...

Question

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

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None of these

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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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