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When arranging individuals around a circular table, the concept of circular permutations comes into play. Unlike a linear arrangement, a circular arrangement considers rotations as identical, meaning the first position is not fixed, and therefore, the number of unique permutations is reduced.
Understanding Circular Permutations:
n
individuals, the number of ways to arrange them in a line is n!
.n
times, making n
linear arrangements identical.(n-1)!
.Applying to the Problem:
6
individuals, the number of circular permutations is (6-1)! = 5! = 120
.Arranging by Birthdates:
2
favorable outcomes.Calculating Probability:
The probability P
of arranging the individuals in the order of their birthdates is:
P=Total Possible ArrangementsNumber of Favorable Outcomes=1202=601
Calculating Odds:
Odds are defined as the ratio of the probability of the event occurring to the probability of it not occurring.
Probability of not arranging in birthdate order is 1 - P = 1 - \frac{1}{60} = \frac{59}{60}
.
Therefore, the odds are:
Odds=1−PP=6059601=591
n!
to (n-1)!
is crucial in circular arrangements because any rotation of the circle results in an identical arrangement.This approach provides a comprehensive understanding of how to calculate the probability and odds of arranging individuals by birthdays in a circular setting, which is a common type of problem in combinatorial mathematics.