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

Given,

${\left(\frac{4x}{5}+\frac{5}{2x^2}\right)}^9$

Now $r^{th}$ term of the expression is given by,

$T_{r+1}={}^{9}C_r{\left(\frac{4x}{5}\right)}^{9-r}{\left(\frac{5}{2x^2}\right)}^r$

$\Rightarrow T_{r+1}={}^{9}C_r{\left(\frac{4}{5}\right)}^{9-r}{\left(\frac{5}{2}\right)}^r{x}^{9-r-2r}$

Now equating $9-r-2r=-6\Rightarrow r=5$

So, coefficient of $x^{-6}$ will be,

$={}^{9}C_5{\left(\frac{4}{5}\right)}^{9-5}{\left(\frac{5}{2}\right)}^5$

$=\frac{9\times 8\times 7\times 6}{4\times 3\times 2\times 1}\times {\left(\frac{4}{5}\right)}^4{\left(\frac{5}{2}\right)}^5$

$=5040$