The neutrino-nucleus cross section and the muon-capture rate are discussed within a simple formalism that facilitates nuclear structure calculations. The corresponding formulas depend on only four types of nuclear matrix elements currently used in nuclear β decay. We have also considered nonlocality effects arising from the velocity-dependent terms in the hadronic current. We show that for both observables in ¹²C the higher order relativistic corrections are of the order of ∼5% only and therefore do not play a significant role. As a nuclear model framework we use the projected quasiparticle random-phase approximation and show that the number projection plays a crucial role in removing the degeneracy between the proton-neutron two-quasiparticle states at the level of the mean field. Comparison is done with both the experimental data and the previous shell model calculations. The possible consequences of the present study on the determination of the ν_μ → ν_e neutrino oscillation probability are briefly addressed.Facultad de Ciencias ExactasInstituto de Física La PlataFacultad de Ciencias Astronómicas y Geofísicas