Now this can be simplified as a product of two individual

parts of a partition function, qA and qB,

which are basically the single particle partition function for the two particles.

Now generally, for the system with N non-interacting

distinguishable particles, each with a single particle partition function Q,

the overall partition function Q is simply giving us small q raised to the power N.

Now, what about indistinguishable particles?

Now, many systems are composed of particles

that are indistinguishable from each other.

For example, particles in the gas phase cannot be distinguished from each other.

Now, for the system of N non-interacting indistinguishable particles,

the partition function summation contains instances where

particles exist in different states, but cannot be distinguished.

Now let's consider one extreme case where all particles exist in different states.

In this case the overcounting factor is N factorial,

which is simply the number of ways of reassigning N particle labels.