
Watch the
Combinations of
n
Different Items
video to explore further the connection between permutations and combinations.

In the Combinations of n Different Items video, you saw a brief introduction to the combination formula. Review it here.
There are various notations, read n choose r or n select r , used to represent a combination.
The first notation is most common and is used exclusively throughout this lesson. You will see both the first two notations in your textbook and the others in Instant Replay videos later in the lesson.
The number of combinations from a set of n different objects, where only r of them are used in each grouping is found using this formula:
Note that this formula can also be written because ( n  r )! r ! and r !( n  r )! are equivalent.
In Example 1, you learned that only one combination occurs when considering all objects in a group. Use the combination formula to verify this finding. Assume the boys have twelve pieces of fruit that will all go into the basket. This means the number of different objects = n = 12 and the number of objects used = r = 12.
This verifies the combination of all objects in a group is one. For this scenario, the result can be written as _{ 12 } C _{ 12 } = 1. In general, the notation is _{ n } C _{ n } = 1 .
Complete the following reading to see the combination formula used to solve problems that do not use all objects in a group.

Read page 113 Example 2 in your textbook, Principles of Mathematics 12 . Complete the Your Turn question on page 113 for practice solving problems using the combination formula. Click here to verify your answer . 
You can use a calculator to verify your work and evaluate combinations given in the form _{ n } C _{ r } . The next two examples show how to evaluate a combination using a calculator. As you read the examples, complete the steps on your calculator . Contact your teacher if you have any difficulty following these examples with your calculator.