Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 1 / 33C / b

For occupations $W_n=Wp(n)$ of $W$ distinguishable copies, the number of arrangements is the multinomial coefficient $\mathcal{N}=W!/{\prod }_n{W}_n!$. The ensemble entropy is $S_W=k_B\log \mathcal{N}$. Stirling's formula and $\sum W_n=W$ giveDividing by the number of independent copies and taking the large-ensemble limit recovers $S=-k_B{\sum }_{n}p(n)\log p(n)$. Terms with zero occupation are interpreted by the usual $0\log 0=0$ convention.