Question

Constant term in the expansion of ${\left(x-\frac{1}{x}\right)}^{10}$ is
(a) 152
(b) −152
(c) −252
(d) 252

Answer 1

(c) −252

Suppose (r + 1)th term is the constant term in the given expansion.
Then, we have:
$$\begin{gathered} T_{r+1}={}^{10}C_r(x{)}^{10-r}{\left(\frac{-1}{x}\right)}^r \\ ={}^{10}C_r(-1{)}^r{x}^{10-r-r} \\ \mathrm{For}\mathrm{this}\mathrm{term}\mathrm{to}\mathrm{be}\mathrm{constant},\mathrm{we}\mathrm{must}\mathrm{have}: \\ 10-2r=0 \\ \Rightarrow r=5 \\ \therefore \mathrm{Required}\mathrm{term}=-{}^{10}C_5=-252 \end{gathered}$$