We consider an irreducible representation of a semisimple Lie algebra L. When restricted to a semisimple subalgebra K of L, this representation can be reduced with respect to K. We derive a general formula for the multiplicity of a certain irreducible representation of K, which occurs in it. The result is an extension of Kostant's formula for the multiplicity of a weight, where the subalgebra K is the Cartan subalgebra of L. Using Kostant's formula, we write down a set of equations, containing the required multiplicity, completely analogous with the usual formula involving the characters. We rewrite these equations using some properties of the partition function (used in Kostant's formula) and of the Weyl groups. Finally we solve them with the help of an ``orthogonality property.'' We illustrate the applicability by working out two nontrivial examples.