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Q.
The coefficient of $x^{4}$ in the expansion of $\left(\frac{x}{2}-\frac{3}{x^{2}}\right)^{10}$ is.
J & K CETJ & K CET 2005
Solution:
The general term in $\left(\frac{x}{2}-\frac{3}{x^{2}}\right)^{10}$
is $T_{r+1}=(-1)^{r-10} C_{r}\left(\frac{x}{2}\right)^{10-r}\left(\frac{3}{x^{2}}\right)^{r}$
$=(-1)^{r} { }^{10} C_{r} \cdot \frac{3^{r}}{2^{10-r}} \cdot x^{10-r-2 r}$
For coefficient of $x^{4}$,
we have to take $10-3 r=4$
$\Rightarrow 3 r=6 \Rightarrow r=2$
$\therefore $ Coefficient of $x^{4}$ in $\left(\frac{x}{2}-\frac{3}{x^{2}}\right)^{10}$
$=(-1)^{2} \cdot{ }^{10} C_{2} \cdot \frac{3^{2}}{2^{8}}=\frac{9 \times 45}{256}=\frac{405}{256}$