# Fundamental Counting Principle: Definition & Examples

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Lesson Transcript
Instructor: Kimberlee Davison

Kim has a Ph.D. in Education and has taught math courses at four colleges, in addition to teaching math to K-12 students in a variety of settings.

In this lesson, you will learn about the fundamental counting principle, a method for determining how many ways choices can be made from groups. Several examples will be given.

## Definition

The fundamental counting principle is a mathematical rule that allows you to find the number of ways that a combination of events can occur. For example, if the first event can occur 3 ways, the second event can occur 4 ways, and the third event can occur 5 ways, then you can find out the number of unique combinations by multiplying: 3 * 4 * 5 = 60 unique combinations.

Imagine that you have a necktie sewing business. You can make unique ties by changing any of the following factors: color (5 options) and shape (3 options). How many unique ties can you make? One way to think about it is by making a diagram. There are 5 colors. Each of the 5 colors can be made into 3 shapes - blue with 3 shape choices, red with 3 shape choices, etc.

By multiplying, you get the total number of paths that you can take through the diagram. You can make 15 different kinds of ties (5 * 3).

Now suppose that you also add 3 pattern choices to your tie options: striped, solid, or polka-dot. How many ties can you make now? Simply imagine one of the possibilities you had originally - maybe a green tie that is short and fat. That green short tie can now be made three ways: striped, solid, or polka-dot. The same is true of the other 14 original ties. So, now you have 15 * 3 = 45 different types of ties.

This multiplication method works any time you have several factors (color, shape, and design) and each of those factors can be combined with each other in any way possible. You can use the fundamental counting rule (multiplication) any time you have a set of categories and one out of several choices in each category will be selected. You might think of it as having several empty 'slots' to fill. Each 'slot' gets only one item.

## A Few Examples

Suppose the slots represent courses in a meal you're going to order. If there are 6 courses, you might have 3 appetizer choices, 2 soup choices, and 4 salad choices, along with 5 main course choices, 10 beverage choices, and 3 dessert choices. To find out how many unique 6-course meals you can make, fill in the blanks with the number of choices and multiply:

3 * 2 * 4 * 5 * 10 * 3 = 3,600 possible unique meals

Another situation might be the creation of license plates. Again, you have 6 slots to fill. This time, the first two slots must be letters (26 choices) and the remaining 4 slots must be numbers (10 choices each). If you fill in the 6 'slots' with the number of choices and multiply, you get the number of license plates you can make:

26 * 26 * 10 * 10 * 10 * 10 = 6,760,000 license plates

## The Fundamental Counting Rule and Exponents

If you have repeats, the same number of choices in several slots, then it's a little more concise to use exponents. The license plate multiplication can be rewritten as:

26^2 * 10^4

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