The microstructure of advanced high-strength steels often shows a sensitive dependence on alloying. For example, adding Cr to improve the corrosion resistance of medium-Mn steels also enhances the precipitation of carbides. The current study focuses on the behavior of H in such complex multicomponent carbides by employing different methodological strategies. We systematically analyze the impact of Cr, Mn, and Fe using density functional theory (DFT) for two prototype precipitate phases, M3C and M23C6, where M represents the metal sublattice. Our results show that the addition of these alloying elements yields strong nonmonotonic chemical trends for the H solubility. We identify magnetovolume effects as the origin for this behavior, which depend on the considered system, the sites occupied by H, and short- vs long-range interactions between H and the alloying elements. We further show that the H solubility is directly correlated with the occupation of its nearest-neighbor shells by Cr and Mn. Based on these insights, DFT data from H containing binary-metal carbides are used to design a ridge regression based model that predicts the solubility of H in the ternary-metal carbides (Fe-Cr-Mn-C).