How does the N signify the indistinguishability in the Gibbs Paradox?

[annaphys]

Gibbs introduced the N! to then make S extensive. He then attributed the N! to the particles being indistinguishable. How does the N! signify the indistinguishability?

 

[BvU]

You can distribute N distinguishable particles over N 'locations' in N! ways.
You can distribute N indistinguishable particles over N 'locations' in one way only.

 

[annaphys]

Nice. Thanks. That makes sense since in deriving the Entropy, we divide by N!. A side question. Are we losing any information of the system by doing this? Of course in quantum mechanics particles are indistinguishable but it'd be interesting to know if any information is lost.

 

[A. Neumaier]

Are we losing any information of the system by doing this? Of course in quantum mechanics particles are indistinguishable but it'd be interesting to know if any information is lost.

Lost is only spurious information that is physically nonexistent anyway.