Question

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

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

Answer 1

The correct option is A

$\frac{405}{256}$

The explanation for the correct option:

Finding the Coefficient of $x^4$

Using permutation and combination of the equation to simplify the equation:

therefore in general term $x^{4}in{[(\frac{x}{2})-(\frac{3}{x^2})]}^{10}=-1^{2.10}=\frac{45\times 9}{256}=\frac{405}{256}$will have value$t_{r+1}=-1$

$=-1^{r10}{C}_r\frac{3^r}{2^{10-r}}.x^{10-3r}$

Put $$\begin{gathered} 10-3r=4 \\ \Rightarrow 3r=6 \\ \Rightarrow r=2 \end{gathered}$$

Hence coefficient of
$$\begin{gathered} x^{4}in{(\frac{x}{2}-\frac{3}{x^2})}^{10}=-1^{2.10} \\ =\frac{45\times 9}{256} \\ =\frac{405}{256} \end{gathered}$$

Hence option A is correct