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Define $q=N_{+}-N_{-}$, where $N_{+}$ counts particles in the $+ϵ$ level. Then $N_{\pm }=(N\pm q)/2$, soThe binomial entropy of independent two-level particles follows from counting which of the $N$ independently labelled particles occupy each nondegenerate level. This is the usual distinguishable, or localized, two-level model; occupation constraints of a degenerate quantum gas would require different counting.

For large occupations, Stirling's approximation with $x=q/N$ givesAllowed $q$ runs from $-N$ to $N$ in steps of two. Entropy is largest near equal populations and is zero when all particles occupy one level; use $0\ln 0=0$ at the endpoints.