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