This thesis presents a study of B->ccbar gamma K decays where ccbar includes J/psi and psi(2S), and K includes K /-, K_S, and K*(892). The particular emphasis is on a search for the radiative decays X(3872)->J/psi gamma and X(3872)->psi(2S) gamma. 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->chi_c1,2 K, which are used as verification for the analysis technique. Show Partial Abstract