  We study lifetimes and weak decays of the triply heavy tetraquarks ${b\bar
c}{b\bar q}$. Following the heavy quark expanding(HQE), the lifetimes of
tetraquarks ${b\bar c}{b\bar q}$ can be expressed as the summation of different
dimension operators. Particularly, we obtain the lifetimes of ${b\bar c}{b\bar
q}$ at the next-to-leading order(NLO) given as $\tau(T^{\{bb\}}_{\{\bar c\bar
q\}})=0.70\times 10^{-12}s$. Besides, we construct the weak decays Hamiltonian
of ${b\bar c}{b\bar q}$ in hadronic level under the SU(3) flavor symmetry. The
discussion of the Hamiltonian can deduce the decay amplitudes and width
relations of the tetraquarks. Following the choosing rules, we collect some
golden channels for the mesonic decays of tetraquarks, which will be helpful to
search for triply heavy tetraquarks ${b\bar c}{b\bar q}$ in future experiments.
