First slide

Binomial theorem for positive integral Index

Question

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

Moderate

A

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

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

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

$(n - 1)$

Solution

We have,
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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