  After the discovery of the double-charm baryon $\Xi_{cc}^{++}$ by LHCb, one
of the most important topics is to search for the bottom-charm baryons which
contain a $b$ quark, a $c$ quark and a light quark. In this work, we study the
two-body non-leptonic weak decays of a bottom-charm baryon into a spin-$1/2$
bottomed baryon and a light pseudoscalar meson with the short-distance
contributions calculated under the factorization hypothesis and the
long-distance contributions considering the final-state-interaction effects.
The branching fractions of all fifty-seven decay channels are estimated. The
results indicate that $\Xi_{bc}^+\to\Xi_b^0\pi^+$,
$\Xi_{bc}^{0}\to\Xi_{b}^{-}\pi^+$ and $\Omega_{bc}^0\to\Omega_b^-\pi^+$ decay
modes have relatively large decay rates and thus could be used to
experimentally search for the bottom-charm baryons. The topological diagrams
and the SU(3) symmetry of bottom-charm baryon decays are discussed.
