Question

Let the coefficients of $x ^{-1}$ and $x ^{-3}$ in the expansion of $\left(2 x^{\frac{1}{5}}-\frac{1}{x^{\frac{1}{5}}}\right)^{15}, x>0$, be $m$ and $n$ respectively. If $r$ is a positive integer such $m n^{2}={ }^{15} C _{ r } .2^{ r }$, then the value of $r$ is equal to _______.

Answer 1

$T _{ r +1}=(-1)^{ r } \cdot{ }^{15} C _{ r } \cdot 2^{15- r } \times \frac{15-2 r }{5}$
$m ={ }^{15} C _{10} 2^{5}$
$n =-1$
so $mn ^{2}={ }^{15} C _{5} 2^{5}$