Question

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

Answer 1

Answer: 5040

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