I have been reading about the Gibbs paradox, in which the assumption that particles of a monoatomic ideal gas are distinguishable leads to a paradox in which entropy is not extensive. In Schroeder's Thermal Physics textbook, this is corrected by assuming that ideal gas particles are indistinguishable, and so the multiplicity function is divided by N! (N = number of particles).

That makes sense to me, however my confusion arises when applying this logic to different systems, for example the Einstein solid. In the textbook, the formula for the entropy of an Einstein solid is S = Nkln[(q/N)+1] where q is the number of energy units shared among the oscillators and N is the number of oscillators. This is extensive since if you replace q --> 2q and N --> 2N, the entropy doubles exactly. But, this formula is derived assuming particles of the Einstein solid are distinguishable, while on the other hand the ideal gas entropy assumes particles are indistinguishable.

My questions are:

1. Why is it correct to assume ideal gas particles are indistinguishable, but not correct to assume oscillators in the Einstein solid are indistinguishable? What specifically results in this difference between the two systems, or is there a general principle that can describe when particles should be considered indistinguishable?
2. How does this notion of indistinguishability play into different definitions of entropy? Specifically, I have read the following derivation of the Gibbs Entropy Formula (S = -k∑p_iln(p_i) where the sum runs over all states i and p_i is the probability of the particle being in the ith state). This derivation also seems to assume distinguishability of particles, since the multiplicity function it is derived from is just a multinomial coefficient. So does that mean that this definition of entropy, applied to the monoatomic ideal gas, is incorrect?

I have been reading about the Gibbs paradox, in which the assumption that particles of a monoatomic ideal gas are distinguishable leads to a paradox in which entropy is not extensive. In Schroeder's Thermal Physics textbook, this is corrected by assuming that ideal gas particles are indistinguishable, and so the multiplicity function is divided by $N!$ ($N$ = number of particles).

That makes sense to me, however my confusion arises when applying this logic to different systems, for example the Einstein solid. In the textbook, the formula for the entropy of an Einstein solid is $S = Nk\ln\left(\frac{q}{N}+1\right)$ where $q$ is the number of energy units shared among the oscillators and $N$ is the number of oscillators. This is extensive since if you replace $q \rightarrow 2q$ and $N \rightarrow 2N$, the entropy doubles exactly. But, this formula is derived assuming particles of the Einstein solid are distinguishable, while on the other hand the ideal gas entropy assumes particles are indistinguishable.

My questions are:

1. Why is it correct to assume ideal gas particles are indistinguishable, but not correct to assume oscillators in the Einstein solid are indistinguishable? What specifically results in this difference between the two systems, or is there a general principle that can describe when particles should be considered indistinguishable?
2. How does this notion of indistinguishability play into different definitions of entropy? Specifically, I have read the following derivation of the Gibbs Entropy Formula ($S = -k \sum p_i \ln(p_i)$ where the sum runs over all states $i$ and $p_i$ is the probability of the particle being in the $i$th state). This derivation also seems to assume distinguishability of particles, since the multiplicity function it is derived from is just a multinomial coefficient. So does that mean that this definition of entropy, applied to the monoatomic ideal gas, is incorrect?