# 4. Permutations

- Page ID
- 21225

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The following topics are included in this series of nine videos.

- Intro to Permutations
- Permutations, example 2
- Permutations, example 3
- Permutation Notation and Guidelines
- Permutations, example 4
- Permutations with stages
- Permutations with cases and stages
- Permutations with complements
- Rearrangements of Letters

Example \(\PageIndex{1}\)

Our classroom has 35 seats and 30 students. If any student could sit in any open seat, how many different seating arrangements are possible?

**Solution**

When the first student walks in, she has 35 options of where to sit. The second student then has 34 options remaining. The third has 33, and so on. Therefore, the answer is \(P(35,30)\).

#### PREWORK

- There are 8 runners in a race. In how many ways can gold, silver, and bronze medals be awarded? Please write your answer using permutation notation and also figure out a numerical value.
- An organization consisting of 8 females and 2 males meets to elect a president, secretary, and treasurer. In how many ways can the positions be filled? In how many ways can they be filled if both sexes must be represented?
- How many 10-letter “words” can be made from the word “antarctica”?

#### Solutions

- Since no runner can receive more than 1 medal, repeats are not allowed. Since the order in which the runners finish determines who gets each medal, the order does matter. Therefore, we use a permutation. There are 8 runners and 3 will get medals so the answer is \(P(8,3)=8\cdot 7\cdot 6=336\).
- One person cannot be in two roles (no repeats), and the roles are clearly different (order matters/different roles) so we can use a permutation. Since there are 10 total people and 3 must be selected, the answer is \(P(10,3)\). For the second part of the problem, we will use the complement. The complement of the set of choices in which both sexes must be represented is the set of choices in which only one sex is represented. There are 0 ways in which only males can fill the roles since there are 3 roles and only 2 males. There are \(P(8,3)\) ways in which only females can fill the roles since there are 8 females. Hence the total number of ways in which only one sex is represented is \(P(8,3)\). Therefore, the number of ways in which both sexes can be represented is \(P(10,3)-P(8,3)\).
- There are 10 letters in "antarctica" with 3 a's, 2 t's, and 2 c's. Hence, the answer is \(\frac{10!}{3!2!2!}\).