Question

In the expansion of ${\left(\frac{x}{2}-\frac{3}{x^2}\right)}^{10}$, the coefficient of $x^4$ is

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

Answer 1

Answer: (A)

$\frac{405}{256}$

Given expansion is ${(\frac{x}{2}-\frac{3}{x^2})}^{10}$

$\therefore$ General therm $T_{r+1}{=}^{10}{C}_r{\left(\frac{x}{2}\right)}^{10-r}{\left(\frac{3}{x^2}\right)}^r$

${=}^{10}{C}_r{x}^{12-r-2r}{\left(2\right)}^{10-r}{3}^r{=}^{10}{C}_r{x}^{12-3r}{\left(2\right)}^{10-r}{3}^r$

Since we have to find the coefficient of $x^4.$

$\therefore 10-3r=4\Rightarrow -3r=-6\Rightarrow r=2.$

Now, coefficient of $x^4$ is ${}^{10}{C}_2{\left(2\right)}^8{3}^2=\frac{405}{256}$