  We consider a binary system of small and large spheres of finite size in a
continuous medium interacting via a non-negative potential. We work in the
canonical ensemble and compute upper and lower bound for the free energy at
finite and infinite volume by first integrating over the small spheres and then
treating the effective system of the large ones which now interact via a
multi-body potential. By exploiting the underlying structure of the original
binary system we prove the convergence of the cluster expansion for the latter
system and obtain a sufficient condition which involves the surface of the
large spheres rather than their volume (as it would have been the case in a
direct application of existing methods directly to the binary system). Our
result is valid for the particular case of hard spheres (colloids) for which we
rigorously treat the depletion interaction.
