= Solution
For $N$ distinguishable lattice sites occupied by $N_i$ mutually indistinguishable atoms of each species, the number of configurations is
$$
\Omega=\frac{N!}{\prod_{i=1}^nN_i!},\qquad
N_i=X_iN,\quad\sum_iX_i=1.
$$
Using <Stirling's approximation>,
$$
\ln\Omega\simeq
N\ln N-\sum_iN_i\ln N_i
=-N\sum_iX_i\ln X_i.
$$
For one mole, $N=N_A$ and $k_BN_A=R$, so
$$
\Delta S_{\rm mix}=-R\sum_iX_i\ln X_i.
$$
<Equiatomic configurational entropy> follows when the composition has $X_i=1/n$, giving
$$
\boxed{\Delta S_{\rm mix}=R\ln n.}
$$
The reference is the unmixed configurational arrangement, and the model assumes random site occupation without <short-range order>. It counts <configurational entropy>, not all vibrational, electronic or magnetic <entropy>. A large number of elements does not by itself guarantee a stable single <solid solution>: competing <thermodynamic phases> and the <enthalpy> term also enter the <Gibbs free energy>.
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