Question

Find the coefficient of $x^4$ in the expansion of ${(x/2-{3/x}^2)}^{10}$

Answer 1

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

$\Rightarrow x^{10-r-2r}=x^4$

$\Rightarrow 10-3r=4$

$\Rightarrow r=2$

$T_3={}^{10}{C}_2{\left(\frac{x}{2}\right)}^8{\left(\frac{-3}{x^2}\right)}^2$

$=\frac{10\times 9}{2}\times \frac{x^8}{2^8}\times \frac{3^2}{x^4}$

$=\frac{810}{512}=\frac{405}{256}$

$\Rightarrow$ Coefficient of $x^4$.