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The semileptonic decays,$B_{c}{\longrightarrow}{\chi_c}(h_c)+{\ell}+{{\nu}}_{\ell}$, and the two-bodynonleptonic decays, $B_{c}{\longrightarrow}{\chi_c}(h_c)+h$, (here $\chi_c$ and$h_c$ denote $(c\bar c[^3P_J])$ and $(c\bar c[^1P_1])$ respectively, and $h$indicates a meson) were computed. All of the form factors appearing in therelevant weak-current matrix elements with $B_c$ as its initial state and a$P$-wave charmonium state as its final state for the decays were preciselyformulated in terms of two independent overlapping-integrations of thewave-functions of $B_c$ and the $P$-wave charmonium and with proper kinematicsfactors being `accompanied'. We found that the decays are quite sizable, sothey may be accessible in Run-II at Tevatron and in the foreseen future at LHC,particularly, when BTeV and LHCB, the special detectors for B-physics, areborne in mind. In addition, we also pointed out that the decays $B_c\toh_c+...$ may potentially be used as a fresh window to look for the $h_c$charmonium state, and the cascade decays, $B_c\to \chi_c[^3P_{1,2}]+l+\nu_l$($B_c\to \chi_c[^3P_{1,2}]+h$) with one of the radiative decays$\chi_c[^3P_{1,2}] \to J/\psi+\gamma$ being followed accordingly, may affectthe observations of $B_c$ meson through the decays $B_{c}\to{J/\psi}+{l}+\nu_{l}$ ($B_c\to J/\psi+h$) substantially.Comment: 24 pages, 3 figures, the replacement for improving the presentation  and adding reference

physical review. d. particles, fields, gravitation, and cosmology/physical review. d. particles and fields

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Decays of the mesonBcto aP-wave charmonium stateχcor