Abstract: The enumeration of combinations with prescribed number of successions of specified length and a related occupancy problem are considered. The generating functions for both situations are derived and the relation between the linear and circular arrangements is obtained. The results are applied to a variety of well known problems, including Kaplansky's ``Problème des Ménages'', the study of Fibonacci and Lucas numbers and the reliability of consecutive k-out-of-n systems.
Key words and phrases: Run, Fibonacci numbers, Lucas numbers, problème des ménages, consecut ive k-out-of-n system.