The speed of sound in an ideal gas is given by

$$ v = \sqrt{\gamma \frac{P}{\rho}}, $$

where:

$$\gamma$$ is the ratio of specific heats,

$$P$$ is the pressure, and

$$\rho$$ is the density.

Since the problem states that $$\frac{P}{\rho}$$ is the same for all the gases, the speed of sound in each gas is determined solely by the factor $$\sqrt{\gamma}$$.

Let's determine the appropriate $$\gamma$$ for each gas:

Helium (He):

Helium is a monatomic gas.

For a monatomic gas, $$\gamma = \frac{5}{3}$$.

Therefore, $$v_{\mathrm{He}} \propto \sqrt{\frac{5}{3}}$$.

Methane (CH$_4$):

Methane is a polyatomic gas (a tetrahedral molecule with three rotational degrees of freedom).

For nonlinear polyatomic gases, $$\gamma$$ is typically taken as $$\frac{4}{3}$$.

Thus, $$v_{\mathrm{CH_4}} \propto \sqrt{\frac{4}{3}}$$.

Carbon Dioxide (CO$_2$):

Carbon dioxide is a linear molecule.

For linear molecules, $$\gamma = \frac{7}{5}$$.

Hence, $$v_{\mathrm{CO_2}} \propto \sqrt{\frac{7}{5}}$$.

Therefore, the ratio of the speeds of sound in the gases is:

$$ v_{\mathrm{He}} : v_{\mathrm{CH_4}} : v_{\mathrm{CO_2}} = \sqrt{\frac{5}{3}} : \sqrt{\frac{4}{3}} : \sqrt{\frac{7}{5}}. $$

Comparing this expression with the given options, we see that it matches Option C.