Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 9B / a

For bosons the occupation number is $n_{\mathbf{p}}=0,1,2,\dots$. Summing the geometric series for the single-mode grand canonical ensemble gives the Bose-Einstein distribution

For fermions the Pauli exclusion principle permits only $n_{\mathbf{p}}=0,1$. The two-term partition sum gives the Fermi-Dirac distribution