Some Combinatorial Identities on the Stirling Number Based on a Probabilistic Model

Abstract

An urn contains \(m\) distinguishable balls with \(m\) distinguishable colors. Balls are drawn for \(n\) times successively at random
and with replacement from the urn. The mathematical expectation of the number of drawn colors is investigated. Some combinatorial identities on the Stirling number of the second kind \(S(n,m)\) are derived by using probabilistic method.