  We study the transition probabilities for the totally asymmetric simple
exclusion process (TASEP) on the infinite integer lattice with a finite, but
arbitrary number of first and second class particles. Using the Bethe ansatz we
present an explicit expression of these quantities in terms of the Bethe wave
function. In a next step it is proved rigorously that this expression can be
written in a compact determinantal form for the case where the order of the
first and second class particles does not change in time. An independent
geometrical approach provides insight into these results and enables us to
generalize the determinantal solution to the multi-class TASEP.
