= Solution
Let $\varepsilon_s$ be the energy of one adsorbed atom, using the same energy reference as the gas-phase value $\varepsilon_0$. For equivalent, noninteracting <adsorption sites>, choosing the occupied positions gives
$$
\Omega(M,N_1)=\binom{M}{N_1}=\frac{M!}{N_1!(M-N_1)!}.
$$
Every configuration has energy $N_1\varepsilon_s$, so the <canonical lattice-gas partition function> is
$$
\boxed{Z_{N_1}=\binom{M}{N_1}e^{-\beta N_1\varepsilon_s}},\qquad \beta=\frac1{k_BT}.
$$
There is no further $1/N_1!$ factor: the combinatorial choice already counts indistinguishable adsorbates correctly. The <Helmholtz free energy>, using <Stirling's approximation> for a macroscopic number of occupied and empty sites, is
$$
A_s\simeq N_1\varepsilon_s+k_BT\left[N_1\ln N_1+(M-N_1)\ln(M-N_1)-M\ln M\right].
$$
Hence the adsorbed-phase <chemical potential> at fixed $T,M$ is
$$
\mu_s=\left(\frac{\partial A_s}{\partial N_1}\right)_{T,M}
=\varepsilon_s+k_BT\ln\frac{N_1}{M-N_1}
=\varepsilon_s+k_BT\ln\frac{\theta_1}{1-\theta_1}.
$$
The logarithm comes from configurational entropy: adding an atom is less favorable when vacancies are scarce. In the gas, the stipulated expression can be written
$$
\mu_g=\varepsilon_0+k_BT\ln\frac{p\Lambda^3}{k_BT},\qquad
\Lambda=\frac{h}{\sqrt{2\pi m k_BT}},
$$
where $\Lambda$ is the <thermal de Broglie wavelength>. Equality of <chemical potentials> at <chemical equilibrium> gives
$$
\frac{\theta_1}{1-\theta_1}=\frac{p\Lambda^3}{k_BT}
e^{(\varepsilon_0-\varepsilon_s)/(k_BT)}=bp.
$$
Therefore
$$
\boxed{\theta_1=\frac{bp}{1+bp}},\qquad
\boxed{b=\frac{\Lambda^3}{k_BT}e^{(\varepsilon_0-\varepsilon_s)/(k_BT)}}.
$$
The <binding energy> $E_b=\varepsilon_0-\varepsilon_s$ is positive for favorable binding. The units of $b$ are inverse pressure, making $bp$ dimensionless. Choosing the adsorbed energy as zero simply sets $\varepsilon_s=0$ and leaves the same physics; it must not remove the relative <binding energy>. This is the <Langmuir isotherm>, with low-pressure linear uptake and high-pressure monolayer saturation.
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