Question

Find the coefficient of $x^{-2}$ in the expansion of $(x^2+\frac{4}{x^5}{)}^6$.

Answer 1

In the expansion $(a{x}^p+\frac{b}{x^q}{)}^n$, the term containing $x^k$ is $T_{r+1}$ where $r=\frac{np-k}{p+q}$.

Here, $r=\frac{12+2}{5+2}=2$.

$T_{2+1}=T_3={6_C}_2\left(x^2{)}^4\right(\frac{4}{x^5}{)}^2$

$={6_C}_2\left(16\right)x^{-2}$

$=\left(15\right)\left(16\right)x^{-2}$

Therefore, required coefficient = 240.