A lattice density functional theory for the nonrandom energy for multicomponent mixtures containing monomers with directional interactions is presented. This theory is a simplification and generalization of a lattice density functional theory developed by Aranovich and Donohue (AD) for two-dimensional and three-dimensional mixtures based upon ideas originally proposed for one-dimensional systems by Ono and Kondo. While quite accurate and general, the AD equations could not be integrated analytically to give expressions for the free energy. With an algebraic rearrangement of this model into a sum of a random mixing internal energy and the deviations from random mixing, an expression is obtained that is both accurate and integrable. Comparisons with Monte Carlo simulations confirm the accuracy of the theory. Unusual phase stability boundaries are predicted.