The ratio of coefficient of $x^{15}$  to the constant term in the expansion of ${\left(x^2+\frac{2}{x}\right)}^{15}$  is

Given  expansion is  ${\left(x^2+\frac{2}{x}\right)}^{15}$ 

 We have general term in the expansion ${\left(x+a\right)}^n$

$(\therefore T_{r+1}={}^n{\text{C}}_r{x}^{n-r}{\left(a\right)}^r$be the expansion of ${(x+a)}^n)$  

General term  $T_{r+1}={}^{15}{\text{C}}_r{\left(x^2\right)}^{15-r}{\left(\frac{2}{x}\right)}^r$

$T_{r+1}={}^{15}{\text{C}}_r{\left(x\right)}^{30-2r}{\left(2\right)}^r{\left(x^{-1}\right)}^r$

       $T_{r+1}={}^{15}{\text{C}}_r{x}^{30-3r}{2}^r$

Compare $x^{30-3r}$  with $x^{15}$

$\Rightarrow x^{30-3r}=x^{15}$

This contains $x^{15}$  if $30-3r=15$  i.e. if $r=5$

$\therefore$   Coefficient of $x^{15}$ $={}^{15}{\text{C}}_5{2}^5=\alpha$  (say)

Also, $T_{r+1}$  is void of $x$  if $\Rightarrow x^{30-3r}=x^0$

( the constant term in the expansion compare with $x^0$ )

$30-3r=0$  i.e. if $r=10$

$\therefore$  Constant term $={}^{15}{\text{C}}_{10}{2}^{10}=b$  (say)

Hence, the required ratio $=\frac{a}{b}=\frac{{ }^{15}{\text{C}}_5{2}^5}{{ }^{15}{\text{C}}_{10}{2}^{10}}$

                                               $=\frac{1}{2^5}=\frac{1}{32}$ .                 $\left(\because {}^{15}{\text{C}}_5={}^{15}{\text{C}}_{10}\right)$