Question

In the expansion of ${\left(x-\frac{1}{x}\right)}^6$ , the constant term is

• A. $20$
• B. $-20$
• C. $30$
• D. $-30$

Answer 1

Answer: (B)

The $(r+1)$ th term of an expansion of
$(x+a{)}^n{=}^n{C}_r{x}^r{a}^{n-r}$
Let $(r+1{)}^{th}$ term be the constant term in the expansion of $(x-\frac{1}{x}{)}^6\cdot$
$\therefore T_{r+1}{=}^6{C}_r{x}^r(-\frac{1}{x}{)}^{6-r}$
$=(-1{)}^{6-r}{\cdot }^6{C}_r{x}^r{x}^{-6+r}$
$=(-1{)}^{6-r}{\cdot }^6{C}_r{x}^{2r-6}$
Since, this term is a constant term
$\therefore 2r-6=0$
$\Rightarrow r=3$
$\therefore T_4=(-1{)}^{-3}{}^6{C}_3$
$=-20$