Question

The coefficient of $x^4$ in $(\frac{x}{2}-\frac{3}{x^2}{)}^{10}$ is

• A. $\frac{405}{256}$
• B. $\frac{504}{259}$
• C. $\frac{450}{263}$
• D. None of these

Answer 1

Answer: (A)

The general term in $(\frac{x}{2}-\frac{3}{x^2}{)}^{10}$ is
$t_{r+1}=(-1{)}^r{}^{10}{C}_r(\frac{x}{2}{)}^{10-r}(\frac{3}{x^2}{)}^r$
$=(-1{)}^r{}^{10}{C}_r\frac{3^r}{2^{10-r}}.x^{10-3r}$
For coefficient of $x^4$ , we put $10-3r=4$
$\Rightarrow 3r=6\Rightarrow r=2$
$\therefore$ Coefficient of $x^4$ in $(\frac{x}{2}-\frac{3}{x^2}{)}^{10}=(-1{)}^2.{}^{10}{C}_2.\frac{3^2}{2^8}$
$=\frac{45\times 9}{256}=\frac{405}{256}$