Problem:
Central High School is competing against Northern High School in a backgammon match. Each school has three players, and the contest rules require that each player play two games against each of the other school's players. The match takes place in six rounds, with three games played simultaneously in each round. In how many different ways can the match be scheduled?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Call the players from Central , and , and call the players from Northern , and . Represent the schedule for each Central player by a string of length six consisting of two each of , and . There are possible strings for player . Assume without loss of generality that the string is . Player B's schedule must be a string with no 's in the first two positions, no 's in the next two, and no 's in the last two. If 's string begins with a and a in either order, the next two letters must be an and a , and the last two must be an and a . Because each pair can be ordered in one of two ways, there are such strings. If 's string begins with or , it must be or , respectively. Hence there are possible schedules for for each of the schedules for , and 's schedule is then determined. The total number of possible schedules is .
The problems on this page are the property of the MAA's American Mathematics Competitions