# Probability

## Combinations


### Learning Objectives

After completing this section, you should be able to:
1. Distinguish between permutation and combination uses.
2. Compute combinations.
3. Apply combinations to solve applications.
In Permutations, we studied permutations, which we use to count the number of ways to generate an ordered list of a given length from a group of objects. An important property of permutations is that the order of the list matters: The results of a race and the selection of club officers are examples of lists where the order is important. In other situations, the order is not important. For example, in most card games where a player receives a hand of cards, the order in which the cards are received is irrelevant; in fact, players often rearrange the cards in a way that helps them keep the cards organized.

### Combinations: When Order Doesn’t Matter

In situations in which the order of a list of objects doesn’t matter, the lists are no longer permutations. Instead, we call them combinations.

### Counting Combinations

Permutations and combinations are certainly related, because they both involve choosing a subset of a large group. Let’s explore that connection, so that we can figure out how to use what we know about permutations to help us count combinations. We’ll take a basic example. How many ways can we select 3 letters from the group A, B, C, D, and E? If order matters, that number is . That’s small enough that we can list them all out in the table below.
Now, let’s look back at that list and color-code it so that groupings of the same 3 letters get the same color, as shown in :
After color-coding, we see that the 60 cells can be seen as 10 groups (colors) of 6. That’s no coincidence! We’ve already seen how to compute the number of permutations using the formula To compute the number of combinations, let’s count them another way using the Multiplication Rule for Counting. We’ll do this in two steps:
Step 1: Choose 3 letters (paying no attention to order). 
Step 2: Put those letters in order.
The number of ways to choose 3 letters from this group of 5 (A, B, C, D, E) is the number of combinations we’re looking for; let’s call that number  (read “the number of combinations of 5 objects taken 3 at a time”). We can see from our chart that this is ten (the number of colors used). We can generalize our findings this way: remember that the number of permutations of  things taken  at a time is . That number is also equal to , and so it must be the case that . Dividing both sides of that equation by  gives us the formula below.

### Check Your Understanding


### Key Terms

1. combination

### Key Concepts

1. Permutations are used to count subsets when order matters; combinations work when order doesn't matter.
2. Combinations can also be computed using factorials.

### Formulas

1. The formula for counting combinations is:
      