It is important that plasma composition is calculated in a manner consistent with statistical mechanics, particularly since the Boltzmann equation is the basis from which transport coefficients and the fluid-dynamic equations are derived. It is shown from statistical mechanical considerations based on the Boltzmann equation and the H-theorem that it is (i) not possible for a plasma to have more than one temperature in equilibrium in the absence of external forces and gradients, and (ii) not possible to draw conclusions about the change in entropy of a plasma in the presence of external forces and gradients. Derivations of the two-temperature Saha equation, and more generally calculations of the composition of a multi-temperature plasma, that are based on entropy maximization are therefore invalid. A thermodynamic derivation of the composition of a multi-temperature plasma that is consistent with the statistical mechanical results is presented. The derivation shows that the equilibrium composition of a plasma can be correctly calculated by minimization of the internal or free energy.