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A Permutation of a set is an ordered combination of its elements. The number of permutations of elements taken at a time is denoted by and is determined by:
the factorial notation is defined as:
Example: A physics olympiad team consists of 4 members. a) In how many ways can all members be arranged in a row for a photo? b) How many ways can the captain and vice-captain be chosen?
a) We can use either the fundamental counting principle or permutations to solve this problem. Let's use the fundamental counting principle:
This is a permutation problem in which we have 4 members and we take 4 at a time, therefore:
In the above equation we used:
b) This is a permutation problem in which we have 4 members and we take 2 at a time:
Example: In how many ways can 5 girls and 4 boys be seated on a bench? a) If there are no restrictions. b) If girls and boys alternate.
a) If there are no restrictions, we have:
b) If boys and girls alternate, we should have a girl on each end:
This permutation problem can be interpreted as having 5 girls and taking 5 at a time and having 4 boys and taking 4 at a time: