Past exam of the thermal and statistical physics course of the University of Cambridge / 2017 / 1 / b / Solution

Repeated letters. There are six positions, with two copies of each of two letters. Counting labeled arrangements and dividing by the permutations of each repeated pair gives

Single-occupancy cells. For $N$ indistinguishable particles, specifying which $N$ of the $M$ distinct cells are occupied completely specifies a configuration. ThusFor distinguishable particles, each chosen occupied set admits $N!$ assignments of particle labels:Equivalently, successively place labeled particles in $M,M-1,\dots ,M-N+1$ available cells. The single-occupancy constraint is essential: this is not unrestricted stars and bars. Both counts give one empty configuration at $N=0$; at $N=M$ they give one unlabeled occupation pattern and $M!$ labeled assignments.