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?