The positioning of nucleosomes, the most abundant nucleoprotein complex in eukaryotes, and their physical properties are strongly influenced by sequence-dependent geometric and elastic properties of the wrapped DNA. At present, the theoretical study of this system is largely limited to numerical computation, and access to entropic contributions has been challenging. In particular, the toolset for the systematic introduction of binding-site-mediated constraints that retains analytical tractability—beyond uniform superhelical wrapping—is currently lacking. Here, we present a mathematical framework for introducing such constraints in the context of the rigid-base-pair model that permits local DNA relaxation and yields closed-form expressions for elastic binding free energies. Importantly, the methodology enables the evaluation of individual contributions to the free energy, including entropy. Benchmarking against numerical solutions obtained via Monte Carlo sampling and validation with experimental data ranging from competitive nucleosome reconstitution to spontaneous nucleosome breathing and force-induced unwrapping demonstrates quantitative fidelity. The approach enables genome-wide evaluation of nucleosome binding affinities and can be readily extended to noncanonical and epigenetically modified DNA or histone cores. While the present study is based on the rigid-base-pair model, the same ideas extend to any rigid-body-based model, opening a route to a broad class of problems in biomolecular modeling.