= Gibbs paradox
{c}
Counting permutations of identical classical <particles> as different states produces a spurious <entropy of mixing> when otherwise identical gases are combined. The dilute <ideal gas> <partition function> $Z_N=Z_1^N/N!$ removes that label multiplicity and gives an extensive <entropy> in the thermodynamic limit. For different species, genuinely distinct composition information still produces a <entropy of mixing>. <quantum statistics> supplies the underlying identical-particle counting.
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