We present a lattice theory fora multicomponent mixture of p distinct polymeric species, each of a prescribed architecture but without any cycles and s monomeric species along with a solvent species, the latter playing the role of a reference species whose amount is controlled not by any activity but by the sum rule of fixed amount of material. The theory is an extension of our previous work on a binary mixture of polymers in bulk or a general mixture next to a surface. The model allows for nearest-neighbor interactions between unlike species. The chemical bondings are allowed to be between monomers (of the same species) that are nearest-neighbor. The resulting theory is obtained by solving the model on a Bethe lattice. The theory has a very simple structure and supersedes random mixing approximation to which it reduces in a special limit, the random mixing approximation limit, see text. We study the behavior of a ternary system numerically and compare it with that of a binary system. We also compare the predictions of our theory with find them to be consistent. However, our theoretical predictions are inconsistent with the conventional Flory-Huggins theory, Thus, our theory is superior to the Flory-Huggins theory. (C) 1997 American Institute of Physics.