Question

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

• A. $\frac{405}{256}$
• B. $\frac{504}{259}$
• C. $\frac{450}{263}$
• D. $Noneofthese$

Answer 1

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

As we know that general term in an expansion ${(a+b)}^n$ is given as-

$T_{r+1}={}^n{C}_r{a}^r{b}^{n-r}$

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

Here, $a=\frac{x}{2},b=\frac{-3}{x^2},n=10$

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

$\Rightarrow T_{r+1}={}^{10}{C}_r{x}^{3r-20}{(-3)}^{10-r}{\left(2\right)}^{-r}$

For $x^4$-

$3r-20=4$

$\Rightarrow r=\frac{20+4}{3}=8$

Therefore,

$T_{8+1}={}^{10}{C}_8{(-3)}^{10-8}{\left(2\right)}^{-8}{x}^4$

Coefficient f $x^4=\left(\frac{10\times 9}{2}\right)\times 9\times \frac{1}{2^8}=45\times 9\times \frac{1}{256}=\frac{405}{256}$