In a regular Markov chain with one absorbing state, for sequences starting from a given state and continuing until the absorbing state is reached, the distribution and moment formulae of the total number of transitions is derived in Section 2, and also its probability generating function (p.g.f.). The joint p.g.f. of the transition frequencies is given in Section 3, from which the p.g.f. of one or more transition frequencies is deduced. In Section 4 some moment formulae associated with these transition frequencies are derived. Section 5 is concerned with inference for such Markov chains, when there are a large number of sequences starting from the same given state.