  We consider the totally asymmetric exclusion process on a ring in discrete
time with the backward-ordered sequential update and particle-dependent hopping
probabilities. Using a combinatorial treatment of the Bethe ansatz, we derive
the determinant expression for the non-stationary probability of transitions
between particle configurations. In the continuous-time limit, we find a
generalization of the recent result, obtained by A. R\'akos and G.M. Sch\"utz
for infinite lattice, to the case of ring geometry.
