 Solution

ID: 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

\frac{6 !}{2 ! 2 !} = 180.

Single-occupancy cells. For

N

N

of the

M

distinct cells are occupied completely specifies a configuration. Thus

W_{indist} = (N M) = \frac{M !}{N ! ( M - N )!} .

For distinguishable particles, each chosen occupied set admits

N!

assignments of particle labels:

W_{dist} = (N M) N! = \frac{M !}{( M - N )!} .

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.

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