Repetitions of chains in urn schemes with and without replacement for urns with balls of several types

We consider the distributions of repetition numbers of chains of a given length in sequences obtained by an urn scheme without replacement for an urn containing balls of several colors. The proofs are based on analogies between the urn schemes without replacement and with replacement. The problems of chain repetitions in urn schemes with replacement have been studied by various authors over the past 50 years, but similar problems for urn schemes without replacement have not been investigated. Exact formulas and two-sided inequalities for the probabilities of chain repetitions and mathematical expectations of the numbers of repetitions are obtained.