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We describe implementation and analysis of a first-principles theory, derived in an earlier work, for the leading terms in an expansion of a Gibbs free energy of a multi-component alloy in terms of order parameters that characterize potential, compositional phases. The theory includes effects of rearranging charge and other electronics from changing atomic occupancies on lattice sites. As well as the rigorous description of atomic short-range order in the homogeneously disordered phase, pairwise interaction parameters suited for atomistic modelling in a multicomponent setting can be calculated. From our study of an indicative series of the Cantor-Wu alloys, NiCo, NiCoCr, NiCoFeCr, and NiCoFeMnCr, we find that the interactions are not approximated well either as pseudobinary or restricted to nearest neighbour range. Our computed order-disorder transition temperatures are low, consistent with experimental observations, and the nature of the ordering is dominated by correlations between Ni, Co, and Cr, while Fe and Mn interact weakly. Further atomistic modelling suggests that there is no true single-phase low-temperature ground state for these multicomponent systems. Instead the single-phase solid solution is kept stable to low temperatures by the large configurational entropy and the Fe, Mn dilution effects. The computationally cost-effectiveness of our method makes it a good candidate for further exploration of the space of multicomponent alloys.