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Fundamental Counting Principle

Reference > Mathematics > Algebra > Counting Principles

The fundamental counting principle is a principle we use to help us determine the number of ways in which events can happen. The fundamental counting principle states that if there are m ways for one event to happen, and n ways for another event to happen, then there are mn ways for both events to happen.

We'll take a simple example: I want to flip a coin twice. How many ways are there for this to happen?

Well, the first event is the first flip, and there are 2 ways for that to happen. The second event is the second flip, and there are 2 ways for that to happen. 2 x 2 = 4, so there are 4 ways for the flips to happen. Does this make sense? Well, the number of outcomes is small, so we ought to be able to list them all to verify that we've got it right.


Yes, it works out to four!

Let's try another example. This time we're going to flip a coin, and then roll a die. There are two ways the coin toss can happen, and six ways the die roll can happen. That means there's a total of 2 x 6 = 12 ways it can happen. Let's list them again:

H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6, which is a total of 12.

We can even expand this to more than two events. Let's suppose I went grocery shopping. I bought Bran Flakes, Corn Flakes, and Oatmeal. I bought skim milk and whole milk. I bought blueberries, strawberries, raspberries, and blackberries.  Assuming that I have cereal, milk, and berries for breakfast, in how many different ways can I have breakfast?

There are 3 ways to choose the cereal, 2 ways to choose the milk, and 4 ways to choose berries. Thus, there are 3 x 2 x 4 = 24 different breakfasts available to me. Go ahead and list them all if you want, to verify that it works. If you do list them, it helps to have abbreviations like we did for the coin and die. For example, the cereals could be B, C, and O, the milk could be S or W. When you get to the berries you would have to be a bit more creative, since blueberries and blackberries both start with B! I might use the letter U for blueberries, since that's the only berry that contains the letter U.

This principle works for independent events. What are independent events? They are events that have no effect on each other. If my choice of cereal doesn't affect my choice of milk or berries, and my choice of milk doesn't affect my choice of berries, then these are independent events.

On the other hand, if I think that bran flakes and blackberries taste disgusting together, then choosing bran flakes will limit my berry choices. How we handle that situation is what we'll talk about in the next section.


If I flip a coin 5 times, how many possible results are there?
If I roll a die twice, how many possible results are there?
If I randomly select a number from 1 to 5 (inclusive) and then flip a coin twice, how many possible results are there?
A restaurant has 4 entrees, 5 appetizers, 8 side dishes, and 10 beverages. How many possible meals are there?
On my burger I can choose to have or not have any of the following options: cheese, onion, lettuce, mustard, ketchup, mayonnaise, pickles. How many different ways are there to have the burger?
I'm going to go to the zoo one day in September, and one day in October. In how many ways can I arrange these trips?
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Counting Dependent EventsCounting Dependent Events

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