Question

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

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

Answer 1

The correct option is B $(-1{)}^n(1+n)$

$(1-x)(1-x{)}^n$
$=(1-x)\left[\right(1+n(-x)+\cdots +{}^n{C}_{n-1}(-x{)}^{n-1}+{}^n{C}_n(-x{)}^n]$

$\therefore$ the coefficient of $x^n$ is
${}^n{C}_n(-1{)}^n-{}^n{C}_{n-1}(-1{)}^{n-1}$
$=(-1{)}^n+(-1{)}^{n}n$
$=(-1{)}^n(1+n)$