Problem:
The Annual Interplanetary Mathematics Examination (AIME) is written by a committee of five Martians, five Venusians, and five Earthlings. At meetings, committee members sit at a round table with chairs numbered from to in clockwise order. Committee rules state that a Martian must occupy chair and an Earthling must occupy chair . Furthermore, no Earthling can sit immediately to the left of a Martian, no Martian can sit immediately to the left of a Venusian, and no Venusian can sit immediately to the left of an Earthling. The number of possible seating arrangements for the committee is . Find .
Solution:
Each acceptable seating arrangement can be specified in two steps. The first step is to assign a planet to each chair according to the committee rules. The second step is to assign an individual from the appropriate planet to each seat. Because the committee members from each planet can be seated in any of ways, the second step can be completed in ways. Thus is the number of ways in which the first step can be completed.
In clockwise order around the table, every group of one or more Martians seated together must be followed by a group of one or more Venusians and then a group of one or more Earthlings. Thus the possible assignments of planets to chairs are in one-to-one correspondence with all sequences of positive integers with and . For each , the number of ordered -tuples with is as are the numbers of ordered -tuples with and with . Hence the number of possible assignments of planets to chairs is
The problems on this page are the property of the MAA's American Mathematics Competitions