Question:
The coefficient of $x^{4}$ in $\left(\frac{x}{2}-\frac{3}{x^{2}}\right)^{10}$ is
(a) $\frac{405}{256}$
(b) $\frac{504}{259}$
(c) $\frac{450}{263}$
(d) none of these
Solution:
(a) $\frac{405}{256}$
Suppose $x^{4}$ occurs at the $(r+1)$ th term in the given expansion.
Then, we have
$T_{r+1}={ }^{10} C_{r}\left(\frac{x}{2}\right)^{10-r}\left(\frac{-3}{2 x^{2}}\right)^{r}$
$=(-1)^{r}{ }^{10} C_{r} \frac{3^{r}}{2^{10-r}} x^{10-r-2 r}$
For this term to contain $x^{4}$, we must have :
$10-3 r=4$
$10-3 r=4$
$\Rightarrow r=2$
$\therefore$ Required coefficient $={ }^{10} C_{2} \frac{3^{2}}{2^{8}}=\frac{10 \times 9 \times 9}{2 \times 2^{8}}=\frac{405}{256}$