Question:
The coefficient of $x^{-3}$ in the expansion of $\left(x-\frac{m}{x}\right)^{11}$ is
(a) $-924 m^{7}$
(b) $-792 m^{5}$
(c) $-792 m^{6}$
(d) $-330 m^{7}$
Solution:
(d) $-330 m^{7}$
Let $x^{-3}$ occur at $(r+1)$ th term in the given expansion.
Then, we have
$T_{r+1}={ }^{11} C_{r} x^{11-r}\left(\frac{-m}{x}\right)^{r}$
$=(-1)^{r} \times{ }^{11} C_{r} m^{r} x^{11-r-r}$
For this term to contain $x^{-3}$, we must have
$11-2 r=-3$
$\Rightarrow r=7$
$\therefore$ Required coefficient $=(-1)^{7}{ }^{11} C_{7} m^{7}$
$=-\frac{11 \times 10 \times 9 \times 8}{4 \times 3 \times 2} m^{7}$
$=-330 m^{7}$