A method of deriving density expansions of correlation functions from density expansions of the excess free energy is based on certain thermodynamic relations between derivatives of the grand partition function and derivatives of the excess free energy. The method also depends on using the most general expansion of the interaction energy into components; it is not assumed that this energy is limited to pairwise components. Expansions are obtained for the correlation functions in multicomponent gases and for the correlation functions for solute species in solutions. The method is also applicable to ionic solutions. A mathematical difficulty limits the systems for which these equations are proved to those in which there is a certain flexibility in treating the model.