= Solution
Particles within each species are indistinguishable, while the two species are physically distinguishable and their numbers are fixed. Independence makes the <canonical partition function> factor by species:
$$
\boxed{Z=\frac{Z_1^{N_A}}{N_A!}\frac{Z_1^{N_B}}{N_B!}
=\frac{Z_1^N}{N_A!N_B!},\qquad N=N_A+N_B.}
$$
Since $Z_1\propto T^{3/2}$ at fixed <volume (thermodynamics)>, $U=-\partial_\beta\ln Z=3Nk_BT/2$. The canonical <entropy> is therefore
$$
\boxed{S=k_B\left[N\ln Z_1-\ln(N_A!)-\ln(N_B!)+\frac32N\right].}
$$
For large species populations, <Stirling's approximation> gives the equivalent <Sackur-Tetrode equation> form
$$
\boxed{S\simeq k_B\sum_{i=A,B}N_i\left[\ln\frac{Z_1}{N_i}+\frac52\right].}
$$
For structureless particles with the same translational <partition function>, $Z_1=V/\lambda^3$, where $\lambda$ is their common <thermal de Broglie wavelength>. More generally the supplied common constant factor in $Z_1$ should be retained in the <entropy>.
Writing $x_i=N_i/N$ separates the total-number contribution from the composition term:
$$
S\simeq Nk_B\left[\ln\frac{Z_1}{N}+\frac52\right]
-Nk_B(x_A\ln x_A+x_B\ln x_B).
$$
The last term is the <entropy of mixing> relative to separated ideal components at the same <temperature> and <pressure>. This is the <entropy of a two-species classical ideal gas>. Artificially labeling otherwise identical <chemical atoms> would not justify dividing by separate factorials or adding a mixing <entropy>. The pure-species <limit> is obtained by $x\ln x\to0$.
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