The first two orders of bridge diagrams for the f-bond expansion and the h-bond expansion are calculated for a binary mixture of Lennard-Jones spheres. The method used follows the Legendre polynomial integration methods outlined in the first two papers of this series. As for the pure fluid cases, the thermodynamic results which follow from these methods are found to be in reasonable agreement with the simulation result. Analysis of the thermodynamic and structural results in comparison to the best current bridge function approximations indicate that accurate descriptions of higher order mixtures will require methods beyond the current mean field treatments which are of utility in simple fluids. The methods given are unfortunately not computationally convenient at highest order; however, the lower order diagrams are both accessible and give reasonable numerical results.