Question

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

• A. $405/226$
• B. $504/289$
• C. $450/263$
• D. None of the above

Answer 1

Answer: (D)

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