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.
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
This simplification is particularly helpful if every slot has the same number of options. For example, suppose you're creating 7-digit phone numbers and you can use any of the ten digits (0 through 9) in each slot. You get 10 to the 7th power possibilities.
The Fundamental Counting Rule and Factorials
Another simplification happens if each slot gets one less choice than the preceding slot. For example, suppose you have 10 books and you want to arrange them in your bookcase. You randomly pick up a book and put it first. The first 'slot' has 10 book choices. But now that you've chosen 1 book, you only have 9 left to choose for the second slot. For the third slot, you only have 8, and so forth until you have only 1 book left.
The number of ways you can arrange your books is:
10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
This can be represented as 10! The exclamation point is a math symbol called a factorial. It means to take the number in front of it and multiply it by the next lowest number, then the next number below that, and so forth until you get to 1.
The fundamental counting principle is simply a way of counting the number of combinations you can create when you are making several choices in a row. In cases where you have repeated values, it can be easier to use exponents. Factorials also come in handy when you are multiplying a set of numbers that starts at 1 and increases by 1 in each 'slot.'
Lesson at a Glance
Using the fundamental counting principle will allow you to find the number of unique ways that a combination of events can occur by simply multiplying the number of options for each event. If you have the same number of choices in several slots, you can also use exponents. Factorials help if each slot gets one less choice than the preceding slot.
After reviewing this lesson, you should be able to
- Apply the fundamental counting principle in applicable situations
- Use exponents to simplify calculations
- Calculate properly if you see a factorial after a number
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