Question

Coefficient of $x^4$ in $(\frac{x}{2}-\frac{3}{x^2}{)}^{10}$ is

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

Answer 1

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

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

General term will be $(-1{)}^r{\cdot }^{10}{C}_r(\frac{x}{2}{)}^r(\frac{3}{x^2}{)}^{10-r}$

$=(-1{)}^r{\cdot }^{10}{C}_r\cdot \frac{x^r}{2^r}\cdot \frac{3^{10-r}}{x^{20-2r}}$

$\Rightarrow x^{r-20+2r}=x^4$

$\Rightarrow 3r-20=4$

$\Rightarrow 3r=24$

$\Rightarrow r=8$

$\therefore$ Co-efficient will be

$(-1{)}^r{\cdot }^{10}{C}_r\cdot \frac{3^{10-r}}{2^r}$

$=(-1{)}^8{\cdot }^{10}{C}_8\cdot \frac{3^2}{2^8}$

$=\frac{10\times 9}{2}\times \frac{9}{256}$

$=\frac{405}{256}$