Question

If the constant term, in binomial expansion of ${\left(2x^r+\frac{1}{x^2}\right)}^{10}$ is $180$ , then $r$ is equal to ______

Answer 1

Answer: 8

${\left(2x^r+\frac{1}{x^2}\right)}^{10}$
General term $={}^{10}{C}_R{\left(2x^2\right)}^{10-R}{x}^{-2R}$
$\Rightarrow 2^{10-R^{10}}{C}_R=180\dots \dots .(1)$
$\&(10-R)r-2R=0$
$r=\frac{2R}{10-R}$
$r=\frac{2(R-10)}{10-R}+\frac{20}{10-R}$
$\Rightarrow r=-2+\frac{20}{10-R}\dots \dots ..(2)$
$R=8$ or $5$ reject equation (1) not satisfied
At $R=8$
$2^{10-R}{}^{10}{C}_R=180\Rightarrow r=8$