Binomial entropy of independent two-level particles ($S=k_B\ln \left(\genfrac{}{}{0ex}{}{N}{N_{+}}\right)$)

For independently labelled particles with nondegenerate levels, choosing which $N_{+}$ particles occupy the upper level gives a binomial multiplicity. With $x=(N_{+}-N_{-})/N$, its large-number entropy is $N{k}_B[\ln 2-(1+x)\ln (1+x)/2-(1-x)\ln (1-x)/2]$. This counting assumes distinguishable or localized degrees of freedom, not indistinguishable particles sharing one pair of quantum orbitals.