We perform direct three-dimensional density functional theory (DFT) calculations of two- and three-body interactions in polymer nanocomposites. The nanoparticles are modeled as hard spheres, immersed in a hard-sphere homopolymer melt of freely jointed chains. The two-particle potential of mean force obtained from the DFT is in near quantitative agreement with the potential of mean force obtained from self-consistent polymer reference interaction site model theory. Three-body interactions among three nanoparticles are found to be significant, such that it is not possible to describe these systems with a polymer-mediated two-body interaction calculated from the potential of mean force.