We study masses, lifetimes and weak decays of the triply heavy tetraquarks b c ¯ b q ¯ . The masses of tetraquark are explored under two different models. Further more, our calculations show a possibility of existence for the stable triply heavy tetraquark b c ¯ b q ¯ with J P = 1 + which should be verified in experiment. Following the heavy quark expanding(HQE), the lifetimes of tetraquarks b c ¯ b q ¯ can be expressed as the summation of different dimension operators. Particularly, we obtain the lifetimes of b c ¯ b q ¯ at the next-to-leading order(NLO) given as τ ( T { c ¯ q ¯ } { b b } ) = 0.70 × 10 - 12 s . Besides, we construct the weak decays Hamiltonian of b c ¯ b 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 c ¯ b q ¯ in future experiments.