Abstract : Feynman Graph counting in Quantum Field Theory (QFT) can be formulated in terms of symmetric groups. This leads to expressions for graph counting and symmetry factors in terms of topological transition amplitudes for strings with a cylinder target, related to two dimensional topological field theory. The details of the interactions in the QFT are encoded in the boundary conditions which specify how the strings wind around circles. The QFTs discussed include scalar field theories and QED, where there is no large gauge group.