Video: Fundamental Counting Principle

In this video, we will learn how to find the number of all possible outcomes in a sample space using the Fundamental Counting Principle.

11:48

Video Transcript

In this lesson, we’ll learn how to find the number of all possible outcomes in a sample space by using the fundamental counting principle, sometimes called the product rule for counting. Let’s begin by considering an example.

Say we have the letters A, B, and C. How many different ways do we have to order these?

Well, we could use something called systematic listing; that is, we just write them all out by using some sort of system. Let’s start with a fairly obvious first option. Let’s start with A-B-C. Then, we’re going to keep A in place, and we’re going to switch the other two letters to get A-C-B. That is a different order, and so that’s the second way of ordering them. There are no other ways to order these with A at the beginning, so we’re now going to put B at the beginning. We could have B-A-C. And if we then hold the B in place and switch the other two letters, we get B-C-A. We can repeat this process also keeping C at the beginning. And then, we see that we’ve run out of ways to order this. And so, if we count these up, we see that there are exactly six ways to order these.

And this can be a nice technique. However, there are a few issues. Firstly, if we have lots of options, say we wanted to order six or seven letters, this process could take a really long time. And secondly, it would be very easy to miss a few options. So, let’s see if we can find another way of doing this. Let’s look at using a tree diagram to find the total number of possible outcomes.

If we think about the first set, then we know that there are three different possible choices. We can choose A, B, and C. Now, let’s imagine we’ve chosen the first letter and we’ve got A. The second letter can now only be B or C. If, however, we’d gone down the second path, the second letter could possibly be A or C. And if we’d gone down the third path or the third branch and chosen a letter C , our second letter could either be A or B. Then, when we move on to the third letter, we see if we’ve already chosen an A and a B, we can now only choose a C. And we can repeat this by drawing one branch off of each of the six previous branches. So, let’s count up the total number of outcomes and see where it’s come from.

There were three branches under our first letter. Then, under the second letter, we had two branches coming off of each of those original three. So, there are three times two branches. Finally, under the third letter, there is one branch coming off of each of the previous ones. So, there’s a total of three times two times one, which is six outcomes. And so, the number of ways of arranging the letters here is three times two times one; it’s six. We simply multiply the number of ways of choosing the first letter by the number of ways of choosing the second by the number of ways of choosing the third. And, in fact, we can generalize this.

The fundamental counting principle, sometimes called the product rule for counting, says that if A and B are two independent events such that A has 𝑚 outcomes and B has 𝑛 outcomes, then the total number of distinct outcomes of the two events together is the product of these; it’s 𝑚 times 𝑛. So, let’s look at an example of the application of this principle.

How many three-different-digit numbers could be formed from the set of digits containing the elements one, two, four, and nine?

Remember the fundamental counting principle, sometimes called the product rule for counting, says that if A is an event that has 𝑚 outcomes and B is an event that has 𝑛 outcomes, the total number of outcomes of A and B together is the product of these. It’s 𝑚 times 𝑛. Now, in fact, we have three possible events here. We have the event which is choosing the first digit, the second event is choosing the second digit, and the third event is choosing the third digit. But the counting principle still holds, so we need to find out the number of outcomes we have for choosing each digit and then multiply these together.

There are four possible digits we can choose from. Those are one, two, four, and nine. And so, it’s quite clear to us that there are four different ways of choosing the first digit. Now, we’re told that they are three-different-digit numbers. So, let’s think about how we choose the second digit. Let’s say, for instance, the first digit we chose was the number one. We can no longer use that. And so, there are three different digits to choose from. The number of ways then of choosing our second digit is three. Similarly, let’s move on to the third digit. We’ve now already chosen two numbers out of the set of four. And so, that means that there can only be two numbers left to choose from.

And so, there are four ways of choosing the first digit, three ways of choosing the second, and two ways of choosing the third. The product rule for counting or the counting principle tells us that we can find the total number of outcomes by multiplying these together. That’s four times three times two, which is equal to 24. There are 24 three-different-digit numbers that could be formed from the set of digits one, two, four, and nine.

The menu at a restaurant is shown below. How many ways are there to choose a two-course meal?

So to choose a two-course meal, we’re going to choose one starter and one main. And so, one way that we have to find the total number of meals that there are is to list out all possible options. But this can be quite a long-winded method. And so, instead, we’re going to recall the counting principle or the product rule for counting. This says that if A is an event that has 𝑚 outcomes and B is an event that has 𝑛 outcomes, the total number of outcomes for the two events together is 𝑚 times 𝑛. And this is why it’s called the product rule for counting. We find the product of the number of outcomes.

We see that there are four possible ways of choosing a starter, and there are three possible ways of choosing a main. This means then that the total number of ways of choosing a two-course meal is the product of the number of outcomes. It’s four times three, which is equal to 12. There are 12 ways to choose a two-course meal.

Let’s consider another example.

A construction company has three active sites. There are 20 different ways to drive from site A to site B. There are 16 ways to drive from site B to site C. In how many ways can we drive from site A to site C visiting site B on the way?

Let’s imagine we have these three construction sites. We’re told that there are 20 ways to get from site A to site B and then 16 ways to get from site B to site C. We need to work out the total number of ways that we can get from site A to site B and then to site C. And to do so, we’re therefore going to recall something called the fundamental counting principle. This says that if A is an event that has 𝑚 outcomes and B is another event that has 𝑛 outcomes, then the total number of outcomes for A and B combined is 𝑚 times 𝑛.

Our first event is driving from site A to site B. So, there’s 20 different ways of doing this. There’s 20 outcomes. Then, our second event is driving from site B to site C, and we know that there are 16 outcomes. And so, the total number of ways that we can drive from site A to site B and then to site C is 20 times 16, which is 320. There are 320 different ways of doing that journey.

How many different ways can we pick a team of one man and one woman from a group of 23 men and 14 women?

This scenario effectively has two events. Event one is choosing one man from a total of 23, and event two is choosing one woman from a total of 14. And so, we’re going to need to recall the fundamental counting principle. This says that if A and B are two independent events such that A has 𝑚 outcomes and B has 𝑛 outcomes, there are a total number of 𝑚 times 𝑛 outcomes of these two events together. And so, the total number of ways that we can pick a team of one man and one woman must be 23 times 14. We can use any method we want really to perform this calculation. Let’s use a column method.

Three times four is 12. So, we put a two in this column and we carry the one. Then, two times four is eight and we add the one to get nine. Next, we’re going to do three times one. But since the one is in the tens column, this is just like doing three times 10, and so we add a zero. Three times one is three and two times one is two. We’re going to add these values. Two add zero is two, nine add three is 12, so we’re going to carry a one, and two add one is three. And so, we see that there are 322 ways to pick a team of one man and one woman from a group of 23 men and 14 women.

We’re going to consider just one final example.

Two spinners are spun. The first spinner is numbered from one to five, and the second spinner is numbered from one to seven. Determine the total number of possible outcomes.

To answer this question, we’re going to recall the fundamental counting principle. This says that if A and B are independent events, that is, the outcome of one does not affect the outcome of the other and if A has 𝑚 possible outcomes and B has 𝑛 possible outcomes, the total number of possible outcomes of the two events together is 𝑚 times 𝑛; it’s the product of these.

Now, our two events are spinning the first spinner and spinning the second spinner. And so, we need to consider the total number of outcomes for each event. The first spinner is numbered from one to five, so there are five different scores that we can receive when we spin that first spinner. Then the second spinner is numbered from one to seven. So, there are seven different outcomes; there are seven different scores that we could get. The product rule for counting or the counting principle says that the total number of possible outcomes is the product of these two numbers. It’s five times seven and of course that is equal to 35. And this means that when we spin these two spinners, we can get a total number of 35 possible outcomes.

We’ll now consider the key points from this lesson. In this video, we learned that the fundamental counting principle, which is sometimes called the product rule for counting, can help us to save time when trying to find a total number of outcomes of more than one event. This says that given two independent events A and B, in other words, the outcome of one doesn’t affect the outcome of the other, where A has 𝑚 outcomes and B has 𝑛 outcomes, the total number of outcomes for these events together is the product of these numbers. It’s 𝑚 times 𝑛.

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