Entropy of Mixing & Gibbs Paradox

Why identical particles don't increase entropy when mixed
ΔS_mix = -nR Σᵢ xᵢ ln xᵢ  (distinguishable)  |  ΔS=0 if identical
N particles 60
Mole fraction x₁ 0.50
Particle type
Speed
The Gibbs paradox: if you mix two volumes of the same gas, entropy appears to increase by ΔS = 2Nk ln 2. But this is wrong! Classical statistical mechanics overcounts states for identical particles by N! — dividing by N! (Boltzmann's correction) resolves the paradox. Only distinguishable particles contribute to mixing entropy. This foreshadowed quantum indistinguishability and is encoded in the N! factor in the partition function. The entropy of mixing formula -R Σ xᵢ ln xᵢ applies only to truly different species.