By Henry's law, 

${\mathrm{p}}_{\mathrm{gas} }\propto {\mathrm{X}}_{\mathrm{gas} }$ (in liquid solution)

and ${\mathrm{p}}_{\mathrm{gas} }={\mathrm{K}}_{\mathrm{H}}\times {\mathrm{\chi }}_{\mathrm{gas} }$

${\mathrm{K}}_{\mathrm{H}}=$ Henry's law constant.

$\therefore$ At a given partial pressure of gas, 

${\mathrm{K}}_{\mathrm{H}}\propto \frac{1}{{\mathrm{\chi }}_{\mathrm{gas}}}$

$\therefore$ The greater the value of ${\mathrm{K}}_{\mathrm{H}}$, the less its solubility.

Here, ${\mathrm{K}}_{\mathrm{H}}$ of $\mathrm{Ar}>{\mathrm{O}}_2>{\mathrm{CO}}_2>{\mathrm{CH}}_4$.

$\therefore$ Solubility of ${\mathrm{CH}}_4>{\mathrm{CO}}_2>{\mathrm{O}}_2>\mathrm{Ar}$.