We analyze the decay processes of ${bf B}_c to {bf B}_n M$ with the $SU(3)_F$ flavor symmetry and spin-dependent amplitudes, where ${bf B}_c({bf B}_n)$ and $M$ are the anti-triplet charmed (octet) baryon and nonet meson states, respectively. In the $SU(3)_F$ approach, it is the first time that the decay rates and up-down asymmetries are fully and systematically studied without neglecting the ${cal O}(overline{15})$ contributions of the color anti-symmetric parts in the effective Hamiltonian. Our results of the up-down asymmetries based on $SU(3)_F$ are quite different from the previous theoretical values in the literature. In particular, we find that the up-down symmetry of $ alpha(Lambda_c^+to Xi^0 K^+)_{SU(3)} = 0.94^{+0.06}_{-0.11}$, which is consistent with the recent experimental data of $0.77pm0.78$ by the BESIII Collaboration, but predicted to be zero in the literature. We also examine the $K_S^0-K_L^0$ asymmetries between the decays of ${bf B}_c to {bf B}_n K_S^0$ and ${bf B}_c to {bf B}_n K_L^0$ with both Cabibbo-allowed and doubly Cabibbo-suppressed transitions.
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