An approach to calculate the triplet structure of a simple liquid, that was proposed some years ago by Barrat, Hansen, and Pastore [Phys. Rev. Lett. 58, 2075 (1987)] has been tested in the binary case. This approach is based on a factorization ansatz for the triplet direct correlation function c((3)); the unknown factor function is determined via the sum rule relating c((3)) and the pair direct correlation function which is the only input information of the system that is required in this formalism. We present an efficient and stable numerical algorithm which solves the six (partly coupled) integral equations for the unknown factor functions. Results are given for the case of a binary hard-sphere mixture and complemented by computer simulation data.