Ligand-receptor interactions are ubiquitous in biology and have become popular in materials in view of their applications to programmable self-assembly. Although complex functionalities often emerge from the simultaneous interaction of more than just two linker molecules, state of the art theoretical frameworks enable the calculation of the free energy only in systems featuring one-to-one ligand/receptor binding. In this Communication, we derive a general formula to calculate the free energy of systems featuring simultaneous direct interaction between an arbitrary number of linkers. To exemplify the potential and generality of our approach, we apply it to the systems recently introduced by Parolini et al. [ACS Nano 10, 2392 (2016)] and Halverson and Tkachenko [J. Chem. Phys. 144, 094903 (2016)], both featuring functionalized Brownian particles interacting via three-linker complexes.