The free-energy concentration expansion method (FCEM), originally derived for the theoretical prediction of surface segregation in the presence of interatomic correlations (short-range order) in binary alloys, is extended to multicomponent alloy nanoclusters, and is applied to unravel the compositional structure of unsupported 309 atom Ni-Cu-Pd cubo-octahedrons. It is a first attempt, to the best of our knowledge, to compute ternary alloy nanocluster site-specific compositions, which are typically inaccessible by current experimental techniques. The calculations predict surface segregation of both solutes Pd and Cu leading to separation of Ni-rich cluster core (the “finite size/matter effect”). Due to the core separation, even relatively low Ni content clusters are expected to be magnetic. The Ni enrichment and associated magnetic order can prevail up to temperatures noticeably higher than in bulk alloys of the same overall composition. Depending on temperature and overall composition, Cu and Pd form orderedlike patterns (“compositional oscillations”) or mix at the wrapping surface. As the overall Cu content in the cluster increases (on the expense of Ni), the two solutes exhibit surface multisite competition, namely, Cu displaces Pd sequentially from vertexes to edges and facets, and finally Pd desegregates from the surface sites into the core. FCEM can be conveniently applied also to calculation of “surfaces” of thermodynamic functions for all ranges of composition above the Gibbs triangle, including the currently defined “cluster mixing functions.” It is shown that sharp compositional-structural changes in Ni-Cu-Pd are reflected distinctly in the cluster mixing entropy surface, and the general shape of calculated free-energy surfaces indicates mixing properties of ternary alloy clusters that differ considerably from the corresponding bulk alloys.