The BABAR Collaboration is a high energy physics experiment located at the Stanford Linear Accelerator Center. The primary goal of the experiment is to study charge and parity violation in the B-meson sector, however the copious production of B mesons decaying to other final states allows for a wide-ranging physics program. In particular, one can access the charmonium system via colour-suppressed b → c decays of the type B → ccK.This thesis presents a study of B → ccγK decays where cc includes J/ψ and ψ(2S), and K includes K ± , K 0 S and K * (892). The particular emphasis is on a search for the radiative decays X(3872) → J/ψ γ and X(3872) → ψ(2S)γ. The X(3872) state is a recently-discovered resonance of undetermined quark composition, speculatively a conventional charmonium state or exotic four-quark di-meson molecule. This research is also sensitive to the well-known radiative charmonium decays B → χ c1,2 K, which are used as verification for the analysis technique.This dissertation sets the best B → χ c1 K branching fraction measurements to date, and sees the first evidence for factorization-suppressed B 0 → χ c2 K * 0 decay at a level of 3.6σ. It also provides evidence for X(3872) → J/ψγ and X(3872) → ψ(2S)γ with 3.6σ and 3.3σ significance, respectively.The product of branching fractions B(B ± → X(3872)K ± ) · B(X(3872) → J/ψγ) = (2.8 ± 0.8(stat.) ± 0.2(syst.)) × 10 −6 and B(B ± → X(3872)K ± ) · B(X(3872) → ψ(2S)γ) = (9.5 ± 2.7(stat.) ± 0.9(syst.)) × 10 −6 are measured.These results improve upon previous X(3872) → J/ψ γ measurements, and represent the first evidence for X(3872) → ψ(2S)γ.