Results from Monte Carlo (MC) simulations are reported for a binary mixture of particles A and B placed at the nodes of a three-dimensional cubic Ising lattice, with pairwise nearest neighbor interactions between species i and j, E-ij. The local composition is studied as a function of composition and of the value of the interchange energy (delta = 2E(AB) - E-AA - E-BB). The MC results are used to assess the accuracy of analytical theories proposed in the literature as a function of the interchange energy. The local composition is shown to obey an exact limiting law at high dilution of one component, thus providing a simple expression for the limiting activity coefficient. The activity coefficients of the species and the entropy per particle are computed from the MC data.