  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 $0.77\pm0.78$ 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.
