We report a new method to compute the Interatomic Coulombic Decay (ICD) widths for large clusters which relies on the combination of the projection-operator formalism of scattering theory and the diatomics-in-molecules approach. The total and partial ICD widths of a cluster are computed from the energies and coupling matrix elements of the atomic and diatomic fragments of the system. The method is applied to the helium trimer and the results are compared to fully ab initio widths. A good agreement between the two sets of data is shown. Limitations of the present method are also discussed.