Repetition is a common part of counting and is very useful in statistics.  Lets first off take a look at this simple example.  You are at the Supermarket and need to buy 9 cases of pop.  On the shelf you see Coke, Diet Coke, Dr. Pepper, Sprint, Pepsi, Diet Pepsi, and Squirt.  This is a total of 7 different types of pop.  Out of pure curiosity, you wonder how many different combinations of pop you could possibly have.  To answer this problem we will use combination with repetition.

Repetition

Before we solve this problem we want to take a look at how repetition works.  Repetition is used with combinations when you are allowed to use two or more of the same elements in a combination.  As in the above problem one of the possible cases may have 3 cases of Coke, 3 Dr. Pepper, 1 Diet Coke, 1 Pepsi, and 1 Diet Pepsi.  Repetition can be defined as:

This can be looked at as the number of ways to put r identical objects into n spots.  n in this case is the number of types of pop on the shelf, and r would be the number of cases you need to buy.  Formally stated, The number of r-combinations from a set with n elements.  Therefore in this case we would have:

Before we continue any further with this problem we will need to take a look at Combinations.

Combinations

A combination of a set of objects is a subset of them or simply a subset of the set with r elements.  This subset is called an r-combination.  The number of ways to choose r objects from n objects is:

This is also sometimes denoted as:

This is also called the "binomial coefficient" which is defined by:

So to finally finish our example problem, from above, we will need the number of combinations of (15,9):

So there we have it, the number of total possible combinations of pop is 455.  Now its time to quiz yourself on the next page to see how well you understand the topic.  Click on the Quiz button below.