Imbalanced beauty-charmed baryons are of great significance to the development of heavy flavor physics. In this work, we study the inclusive weak-annihilation decays of and their contributions to the lifetimes. For the calculation of the inclusive decay width where stands for the sum of the final states with charm number and strange number , we work in the heavy diquark effective theory, which provides us with a convenient technical tool to construct the operator product expansion. The is considered to be a superposition of two states with one containing a spin-0 diquark and the other one containing a spin-1 diquark. It is found that both the lifetimes and the branching ratios are very sensitive to the spin in . The results are also presented. As has a longer lifetime than and bigger branching ratios of similar decay channels, the exclusive decays , , , and are more promising for experimental searches of at the LHC comparing with exclusive decay .