# Probability

## Permutations


### Learning Objectives

After completing this section, you should be able to:
1. Use the Multiplication Rule for Counting to determine the number of permutations.
2. Compute expressions containing factorials.
3. Compute permutations.
4. Apply permutations to solve problems.
Swimming events are some of the most popular events at the summer Olympic Games. In the finals of each event, 8 swimmers compete at the same time, making for some exciting finishes. How many different orders of finish are possible in these events? In this section, we’ll extend the Multiplication Rule for Counting to help answer questions like this one, which relate to permutations. A permutation is an ordered list of objects taken from a given population. The length of the list is given, and the list cannot contain any repeated items. 

### Applying the Multiplication Rule for Counting to Permutations

In the case of the swimming finals, one possible permutation of length 3 would be the list of medal winners (first, second, and third place finishers). A permutation of length 8 would be the full order of finish (first place through eighth place). Let’s use the Multiplication Rule for Counting to figure out how many of each of these permutations there are.

### Factorials

The pattern we see in  occurs commonly enough that we have a name for it: factorial. For any positive whole number , we define the factorial of  (denoted  and read " factorial") to be the product of every whole number less than or equal to . We also define 0! to be equal to one. We will use factorials in a couple of different contexts, so let's get some practice doing computations with them.

### Permutations

As we’ve seen, factorials can pop up when we’re computing permutations. In fact, there is a formula that we can use to make that connection explicit. Let’s define some notation first. If we have a collection of  objects and we wish to create an ordered list of  of the objects (where ), we’ll call the number of those permutations  (read “the number of permutations of  objects taken  at a time”). We formalize the formula we'll use to compute permutations below.
If you wondered why we defined  earlier, it was to make formulas like this one work; if we have  objects and want to order all of them (so, we want the number of permutations of  objects taken  at a time), we get . Next, we’ll get some practice computing these permutations.

### Check Your Understanding


### Key Terms

1. permutation
2. factorial

### Key Concepts

1. Using the Multiplication Rule for Counting to enumerate permutations.
2. Simplifying and computing expressions involving factorials.
3. Using factorials to count permutations.

### Formulas

1. 