Question

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

• A. $(- 1)^{n-1} n $
• B. $(- 1)^n (1 -n)$
• C. $(-1)^{n-1} (n-1)^2$
• D. $(n - 1 ) $

Answer 1

Answer: (B)

Coeff. of $x^{n}$ in $ \left(1+x\right)\left(1-x\right)^{n}$
= coeff of $x^n$ in
$ \left(1+x\right)\left(1 - {^{n}C_{1}}x + {^{n}C_{2}}x^{2} - ...+\left(-1\right)^{n} \times \,{}^nC_{n} x^{n}\right) $
$= \left(-1\right)^{n} \times \,{^{n}C_{n} }+ \left(-1\right)^{n-1}\, {^{n}C_{n-1}} = \left(-1\right)^{n} + \left(-1\right)^{n-1} .n$
$ = \left(-1\right)^{n}\left(1-n\right)$