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We give a geometric method for determining the cohomology groups of apolyhedral product under suitable freeness conditions or with coefficientstaken in a field. This is done by considering first the special case for whichthe pairs of spaces are wedge decomposable. We derive a decomposition for thesepolyhedral products which resembles a Cartan formula. The theory of symmetricproducts is used then to generalize the result to polyhedral products involvingarbitrary pairs. This leads to a direct computation of the Hilbert-Poincar\'eseries and to other applications.

Symmetric Products and a Cartan-type formula for polyhedral products