  We study the semileptonic decays ${\bf B_c} \to {\bf B_n} \ell^+ \nu_{\ell}$
where ${\bf B_{c(n)}}$ is the anti-triplet-charmed (octet) baryon with the
$SU(3)_f$ flavor symmetry and helicity formalism. In particular, we present the
decay branching ratios of $ {\bf B_c} \to {\bf B_n}\ell^+ \nu_\ell$ in three
scenarios: (a) an exact $SU(3)_f$ symmetry with equal masses for the
anti-triplet-charmed (octet) baryon states of ${\bf B_c}$ (${\bf B_n}$), (b)
$SU(3)_f$ parameters without the baryonic momentum-transfer dependence, and (c)
$SU(3)_f$ with baryonic transition form factors in the heavy quark limit. We
show that our results are all consistent with the existing data. Explicitly, we
predict that ${\cal B}( \Xi_c^+ \to \Xi^0 e^+ \nu_{e})=(11.9\pm1.3, 9.8\pm 1.1,
10.7\pm 0.9)\times 10^{-2}$ and ${\cal B}( \Xi_c^0 \to \Xi^- e^+
\nu_{e})=(3.0\pm0.3, 2.4\pm 0.3, 2.7\pm 0.2)\times 10^{-2}$ in the scenarios
(a), (b) and (c) agree with the data of $(14.0^{+8.3}_{-8.7})\times 10^{-2}$
and $(5.6\pm2.6)\times 10^{-2}$ from the CLEO Collaboration, respectively. In
addition, we obtain that ${\cal B}(\Lambda_c^+\to n e^+ \nu_e)=(2.8\pm 0.4,
4.9\pm0.4, 5.1\pm0.4)\times 10^{-3}$ in (a), (b) and (c). We also examine the
longitudinal asymmetry parameters of $\alpha({\bf B_c} \to {\bf B_n} \ell^+
\nu_{\ell})$, which are sensitive to the different scenarios with $SU(3)_f$.
Some of the decay branching ratios and asymmetries can be observed by the
ongoing experiments at BESIII and LHCb as well as the future searches by
BELLEII.
