= Solution
Put $\beta=1/(k_BT)$ and define the <thermal de Broglie wavelength> of species $\alpha$ by
$$
\lambda_\alpha=\frac{h}{\sqrt{2\pi m_\alpha k_BT}}=\sqrt{\frac{2\pi\hbar^2\beta}{m_\alpha}}.
$$
Ignoring the internal <degrees of freedom> as stipulated, each <molecule>'s translational <canonical partition function> is $V/\lambda_\alpha^3$. The two species do not interact, so their <partition functions> multiply; the treatment of <indistinguishable particles> requires a separate factorial for each species, not a factorial of the total <particle> number. Thus
$$
\boxed{Z_{\mathrm{CO-O_2}}=\frac{(V/\lambda_{\rm CO}^3)^{N_{\rm CO}}}{N_{\rm CO}!}\frac{(V/\lambda_{\rm O_2}^3)^{N_{\rm O_2}}}{N_{\rm O_2}!}.}
$$
Using $F=-k_BT\ln Z$ and <Stirling's approximation> gives the extensive <Helmholtz free energy>
$$
\boxed{F_{\mathrm{CO-O_2}}=k_BT\sum_{\alpha=\mathrm{CO,O_2}}N_\alpha\left[\ln\left(\frac{N_\alpha\lambda_\alpha^3}{V}\right)-1\right].}
$$
These are <ideal gas> expressions, valid for $n_\alpha\lambda_\alpha^3\ll1$, with $n_\alpha=N_\alpha/V$. The <energy> zero has been chosen so that separated CO and <oxygen> <molecules> have zero internal reference <energy>.
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