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

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.