University of Georgia
EMAT 6690
Claudette Tucker
Essay Five
InterMath: The Card Game
Investigation: There are five friends who meet each week to play, and they play at a table of six chairs. The diagram below shows that the extra chair is irrelevant to the problem because the placement of the extra chair does not alter the seating of friends.
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Therefore, if we rotated each friend over one to the left or right, we will still have the same seating arrangement. In order to determine, the exact number of times the friends have played cards together, we are interested in the number of different ways of adjusting the seating positions. We can obtain the number of arrangements by performing permutations or specifically circular permutations, which are permutations that occur when objects are assembled in a circle.
Circular Permutation Theorem: The number of permutations of n distant objects arranged in a circle is (n – 1)!.
The reason why we found it necessary to use the above theorem to solve the problem is because we are looking for distinct seating arrangements which will reveal the number of time the friends have played cards together. If we used n! instead of (n -1)! to yield the result, it would duplicate several seating arrangements, such as ABCDE and EDCBA.
To eliminate indistinctive seating arrangements, one friend remains in his or her position while the other four positions are changed continuously. According to the theorem, (n – 1)! = (5 – 1)! = 4! = 4*3*2*1 = 24. So, since the friends have exhausted all possible seating arrangements and given the circular permutation theorem, we can conclude that the five friends have played cards together 24 times.