Question

The coefficient of $x^{-6}$, in the expansion of $\left(\frac{4 x}{5}+\frac{5}{2 x^2}\right)^9$, is______

Answer 1

$\left(\frac{4 x}{5}+\frac{5}{2 x^2}\right)^9$
Now, $T _{ r +1}={ }^9 C _{ r } \cdot\left(\frac{4 x }{5}\right)^{9- r }\left(\frac{5}{2 x ^2}\right)^{ r }$
$={ }^9 C _{ r } \cdot\left(\frac{4}{5}\right)^{9- r }\left(\frac{5}{2}\right)^{ r } \cdot x ^{9-3 r }$
Coefficient of $x^{-6}$ i.e. $9-3 r=-6 \Rightarrow r=5$
So, Coefficient of $x ^{-6}={ }^9 C _5\left(\frac{4}{5}\right)^4 \cdot\left(\frac{5}{2}\right)^5=5040$