  This letter presents a new, solely thermodynamical argument for considering
the states of the quantum isolated horizon of a black hole as distinguishable.
We claim that only if the states are distinguishable, the thermodynamic entropy
is an extensive quantity and can be well-defined. To show this, we make a
comparison with a classical ideal gas system whose statistical description
makes only sense if an additional 1/N!-factor is included in the state counting
in order to cure the Gibbs paradox. The case of the statistical description of
a quantum isolated horizon is elaborated, to make the claim evident.
