The coefficient of x

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 $=(-1)^{7}{ }^{11} C_{7} m^{7}$

$=-\frac{11 \times 10 \times 9 \times 8}{4 \times 3 \times 2} m^{7}$

$=-330 m^{7}$

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