An elementary method is described for deriving the Boltzmann, Fermi–Dirac, and Bose–Einstein distributions that neither uses Lagrange undetermined multipliers nor requires a knowledge of Stirling’s approximation for ln(n!). Apart from giving a more direct account of the optimal properties of these distributions, which may be useful when first introducing statistical thermodynamics, the method also shows clearly how the validity of the distributions depends upon the size of the system. Extensions of the method demonstrate the statistical origin of temperature and the dominance of the most probable macrostate.