How to Use the Fundamental Counting Principle

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  • 0:01 Counting Combinations
  • 1:39 Combination Vocabulary
  • 3:11 Fundamental Counting Principle
  • 4:09 Applying the Principle
  • 5:08 Lesson Summary
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Lesson Transcript
Instructor: Joseph Vigil
There are many situations in which you will have to make several decisions simultaneously. The fundamental counting principle will help you determine how many different possible outcomes there are when you have to make multiple simultaneous decisions.

Counting Combinations

Choices, choices, choices; it seems we're bombarded with choices daily. It can be hard enough making up your mind about a single decision, but what about when you have to make multiple decisions at once?

Restaurants these days, for example, have great lunch specials. Let's say we go to your favorite restaurant for lunch. Their special consists of a drink (soda, tea or lemonade), a salad (garden or Caesar) and a small entree (pasta, chicken or meatloaf). Talk about decisions!

Let's slow down and take a look at all of our options. Just how many different possible lunch combinations are there? Well, let's look at the possibilities. For any drink you choose, you could then go with one of two salads. So each drink option branches out into two further decisions.

From each of those two salad options, you then have three entree options. So those two possibilities now become six. In other words, when we start with one drink, we then have six possible combinations that branch out from there. Since we have three drinks, we can multiply three times six to find the total number of lunch combinations. 3 * 6 = 18, so there are 18 possible lunch combinations.

We've used the fundamental counting principle without even knowing it. But before we examine that principle more closely, let's look at some fancy vocabulary you might find in these situations.

Combination Vocabulary

Remember how you had to make three separate decisions for lunch: drink, salad and entree? Each of these single decisions is called an event. An event, in the world of probability, is a single occurrence or decision with a distinct set of possible outcomes. For example, if you flip a coin, that single coin flip is an event, because it's one occurrence that has a defined set of possible outcomes.

Every event has a sample space, which is simply the complete set of possible outcomes for a single event. For a coin flip, the sample space includes heads and tails because those are the possible outcomes in the event of a coin flip.

And, finally, sample points are the individual possible outcomes in a sample space. So those two possible outcomes in our coin flip - heads and tails - are our two sample points. Some sample spaces are larger than others. It just depends on how many possible outcomes there are in a single event.

For example, if we're determining whether an item from a garden is a fruit or vegetable, that's a small sample space because there are only two possible outcomes, or sample points. But if we then determine what type of fruit or vegetable that item is, our sample space could potentially become much larger because it consists of every kind of fruit and vegetable contained in that garden.

The Fundamental Counting Principle

According to the fundamental counting principle, when we're dealing with multiple events, we can multiply together the number of sample points in each event to determine the total number of possible combinations.

This is essentially what we did when we determined the total number of possible lunch combinations. We had three events: drink choice, salad choice and entree choice.

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