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Q.
The coefficient of $x^4$ in $\bigg(\frac{x}{2}-\frac{3}{x^2} \bigg)^{10}$ is
IIT JEEIIT JEE 1983Binomial Theorem
Solution:
The general term in $\bigg(\frac{x}{2}-\frac{3}{x^2} \bigg)^{10}$ is
$ t_{r+1}=(-1)^r \,{}^{10}C_r \bigg(\frac{x}{2}\bigg)^{10-r}\bigg(\frac{3}{x^2}\bigg)^r $
$=(-1)^r \,{}^{10}C_r \frac{3^r}{2^{10-r}}.x^{10-3r}$
For coefficient of $x^4$ , we put $10 - 3r = 4$
$\Rightarrow 3r=6 \Rightarrow r=2$
$\therefore $ Coefficient of $x^4 $ in $\bigg(\frac{x}{2}-\frac{3}{x^2}\bigg)^{10}=(-1)^2. \,{}^{10}C_2.\frac{3^2}{2^8}$
$ =\frac{45\times 9}{256}=\frac{405}{256}$