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The Boltzmann distribution is one of the most fundamental results in statistical mechanics. We apply this distribution so often that it’s worth thinking about the assumptions involved in deriving it. In this talk, I will attempt to demonstrate that the Boltzmann distribution is a purely statistical result about a very particular combinatorics (i.e. counting) problem. In the instructional section, I will work through two differently-flavoured derivations of the Boltzmann distribution, and discuss how to easily arrive at distribution functions in various statistical ensembles. With this in hand, I will strip away most of the physical complexity; in the “original research” section, I will motivate and solve the combinatorics problem at the heart of the Boltzmann distribution. I’ll also discuss and develop an importance sampling algorithm for efficient numerical solution of this problem.