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In order to derive the partition function for classical particles it is said that one must take into account the indistinguishability of particles by adding $1/N!$ to the partition function (otherwise we can get into trouble because of Gibbs Paradox).

In these notes they argue that it must be added a multinomial coefficient, taking into account the number of possible ways in which $N$ distinguishable particles can be put into individual quantum states such that there are $n_1$ particles in state $1$, $n_2$ in state 2, etc.

So classical particles must be taken as distinguishable or indistinguishable? why in the multinomial coefficient case they take into account quantum states $n_i$ in a classical particles analysis?

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    $\begingroup$ So far as we can tell, the properties (i.e., mass, charge, etc.) of any two electrons are exactly identical. Thus, in the classical sense one cannot "paint" one electron blue and the other red in order to distinguish them. However, in quantum, we can force one into a specific state and the other into a different state (e.g., make one spin up and the other spin down), in which case they can be distinguished. [Probably overly simplified, but hopefully helpful] $\endgroup$
    – honeste_vivere
    Commented Jun 20, 2017 at 16:56

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Classical particles have to be taken as distinguishable. Therefore, the Boltzmann distribution treats the particles as distinguishable.

However, quantum particles are indistinguishable. That came as a big surprise and has several consequences, e.g. Bose-Einstein condensates exist. This contains a nice introduction.

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