The coefficient of xn in the expansion of (1+x)(1−x)n is

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

A
$(- 1)^{n} n$
B
$(- 1)^{n} (1 - n)$
C
$(- 1)^{n - 1} (n - 1)$
D
$(n - 1)$

Solution:

$\begin{array}{l} (1 + x) (1 - x)^{n} = [2 - (1 - x)] (1 - x)^{n} \\ = 2 (1 - x)^{n} - (1 - x)^{n + 1} \end{array}$

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

$\begin{array}{l} = 2 (- 1)^{n} -^{n + 1} C_{n} (- 1)^{n} \\ = (- 1)^{n} (2 - n - 1) = (- 1)^{n} (1 - n) \end{array}$

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