Question

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

• A. $-\frac{405}{256}$
• B. $\frac{405}{256}$
• C. $-\frac{405}{16}$
• D. $\frac{405}{16}$

Answer 1

The correct option is B $\frac{405}{256}$

In the expansion of ${(\frac{x}{2}-\frac{3}{x^2})}^{10}$,
The general term is
$T_{r+1}={}^{10}{C}_r{\left(\frac{x}{2}\right)}^{10-r}{(-\frac{3}{x^2})}^r\Rightarrow T_{r+1}={}^{10}{C}_r(-1{)}^r\frac{3^r}{2^{10-r}}{x}^{10-3r}$

Here, we need the coefficient of $x^4$
$\therefore 10-3r=4\Rightarrow r=2$
$\therefore T_{2+1}={}^{10}{C}_2(-1{)}^2\frac{3^2}{2^{10-2}}{x}^4=\frac{405}{256}{x}^4$
$\therefore$ The coefficient of $x^4$ is $\frac{405}{256}$