Question

The coefficient of $x^{-3}$ in the expansion of ${\left(x-\frac{m}{x}\right)}^{11}$ is
(a) $-924m^7$
(b) $-792m^5$
(c) $-792m^6$
(d) $-330m^7$

Answer 1

(d) $-330m^7$
$$\begin{gathered} \mathrm{Let}{x}^{-3}\mathrm{occur}\mathrm{at}(r+1)\mathrm{th}\mathrm{term}\mathrm{in}\mathrm{the}\mathrm{given}\mathrm{expansion}. \\ \mathrm{Then},\mathrm{we}\mathrm{have} \\ {T}_{r+1}={}^{11}C_r{x}^{11-r}{\left(\frac{-m}{x}\right)}^r \\ =(-1{)}^r\times {}^{11}C_r{m}^r{x}^{11-r-r} \\ \mathrm{For}\mathrm{this}\mathrm{term}\mathrm{to}\mathrm{contain}{x}^{-3},\mathrm{we}\mathrm{must}\mathrm{have} \\ 11-2r=-3 \\ \Rightarrow r=7 \\ \therefore \mathrm{Required}\mathrm{coefficient}=(-1{)}^7{}^{11}C_7{m}^7 \\ =-\frac{11\times 10\times 9\times 8}{4\times 3\times 2}{m}^7 \\ =-330m^7 \end{gathered}$$