A two-dimensional Monte Carlo simulation method based on the NpT ensemble and the Voronoi tesselation, which was previously developed for single-species hard-disk systems, is extended, along with a version of scaled-particle theory, to many-component mixtures. These systems are unusual in the sense that their composition is not fixed, but rather determined by a set of internal degeneracies assigned to the differently sized hard disks, where the larger disks have the higher degeneracies. Such systems are models of monolayers of molecules with internal degrees of freedom. The combined set of translational and internal degrees of freedom leads to a rich phase structure that includes solid-liquid transitions (governed by the translational variables) as well as transitions involving changes in average disk size (governed by the internal variables). The relationship between these two types of transitions is studied by the method in the case of a binary mixture, and results are presented for varying disk-size ratios and degeneracies. The results are also compared with the predictions of the extended scaled-particle theory. Applications of the model are discussed in relation to lipid monolayers spread on air-water interfaces, and it is concluded, by comparison with experiments, that the hard-disk mixture is an excellent candidate for a minimal model of lipid-monolayer phase behavior.