For more particles, we would get lots of terms, the first where all particles were in the same state, the last where all particles are in different states, where the last term must be divided by N! to eliminate overcounts. Thus, we should write
For more particles, we would get lots of terms, the first where allparticles were in the same state, the last where all particles are in different states, where the last term must be divided by N! to eliminate overcounts. Thus, we should write