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**Permutations **

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: