The ratio of the coefficient of x10 in 1+x210 and the term independent of x in x+2x10 is

$T_{r + 1} = (r + 1)$ th term in ${(1 + x^{2})}^{10} =^{10} C_{r} {(x^{2})}^{r}$

$∴ coefficient of x^{10} in {(1 + x^{2})}^{10} {is}^{10} C_{5} .$

Also, $t_{r + 1} = (r + 1)$ th term in ${(x + \frac{2}{x})}^{10}$

$\begin{array}{l} =^{10} C_{r} (x)^{10 - r} {(\frac{2}{x})}^{r} \\ =^{10} C_{r} 2^{r} x^{10 - 2 r} \end{array}$

For this term to be independent of $x,$ set $10 - 2 r = 0$

or $r = 5 .$

$∴$ term independent of $x$ is $^{10} C_{5} (2^{5})$

The ratio of the coefficient of $x^{10}$ in ${(1 + x^{2})}^{10}$ and the term independent of $x$ in ${(x + \frac{2}{x})}^{10}$ is