= Solution
For any nonrelativistic species $j$, part (a) gives
$$
n_j=g_j\left(\frac{2\pi m_jk_BT}{h^2}\right)^{3/2}
e^{(\mu_j-m_jc^2)/(k_BT)}.
$$
Chemical equilibrium and zero photon chemical potential imply
$$
\mu_{H^+}+\mu_e=\mu_{H^0},
\qquad
\mu_{He^+}+\mu_e=\mu_{He^0}.
$$
Taking the appropriate ratios of the number-density formula, approximating the ion and atom masses as equal in the translational prefactors, and using the given degeneracies yields
$$
\boxed{
\frac{n_en_{H^+}}{n_{H^0}}
=\left(\frac{2\pi m_ek_BT}{h^2}\right)^{3/2}
e^{-I_H/(k_BT)}}
$$
and
$$
\boxed{
\frac{n_en_{He^+}}{n_{He^0}}
=2\left(\frac{2\pi m_ek_BT}{h^2}\right)^{3/2}
e^{-I_{He}/(k_BT)}}.
$$
The factors are $g_eg_{H^+}/g_{H^0}=1$ and $g_eg_{He^+}/g_{He^0}=2$.
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