By expanding both the interaction operator and the distribution function (in a quantum mechanical Liouville equation) in power series of a small quantum parameter lambda, semiclassical BBGKY equation have been derived. A general method for its solution have been developed, yielding the various order of quantum corrections to the solutions of the classical BBGKY equation. Using the classical streaming operators in its formulations, these solutions can be interpreted in terms of motions along well-defined classical trajectories.