Explainer: Fundamental Counting Principle

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

Consider a situation where you are buying a new phone; you have two options for size: the 5-inch and the 6-inch models, and there are three color options: black, gold, and white. You would like to know how many options you have in total. One of the easiest ways to represent a situation like this is with a tree diagram. The tree diagram below shows the two options for phone size; then, below each of these we show the three options for color.

We could equally represent these options in a tree diagram where the first choice is the choice of color and the second is the choice of size as shown below.

From this diagram, we can see that there are a total of six options. We could have also got to this answer by listing all the possible options. Of course, drawing a tree diagram or listing all possible options is not practical when we have even a moderate number of options. For example, it would not be practical to draw a tree diagram to find the number of possible outfits that can be made with 5 tops, 5 skirts, and 5 pairs of shoes. We, therefore, need a better method for calculating the number of possibilities.

If we consider what we are doing when we build a tree diagram, we will quickly see how we can generalize this to work with a larger number of options. For the phone example, we started by considering one of the choices, for example, the size of the phone. For this, we have two options; then, for each of these two options we can pick one of the three colors. Hence, the total number of possibilities is 2Γ—3. This method for finding the number of possibilities or outcomes is referred to as the fundamental counting principle.

Definition: The Fundamental Counting Principle

If we have two independent events 𝐴 and 𝐡 such that the number of possible outcomes for event 𝐴 is π‘₯ and the number of possible outcomes for event 𝐡 is 𝑦, the total number of distinct possible outcomes of these two events together is the product π‘₯×𝑦.

In this definition, we used the term independent events. What we mean by this is that the outcome of one event does not change the possible outcomes of the other event. For example, if you were to choose two chocolates from a box containing 4 chocolates, the number of possible outcomes is not 4Γ—4. The reason for this is that when you choose your first chocolate, you change the possible outcomes of the second event; by taking one chocolate, you reduce the number of possible outcomes for your second choice since, now, there are only three chocolates left in the box. In cases where the events affect one another like this, we cannot find the total number of outcomes by simply multiplying the possible outcomes of the separate events as if they happened independently; we need to consider the way the two events impact one another.

Example 1: Applying the Fundamental Counting Principle

A cafe offers a choice of 20 meals and 12 beverages. In how many different ways can a person choose a meal and a beverage?

Answer

Applying the counting principle, we have 20 choices of meals and 12 for beverages, so the total number of ways a person can form a distinct meal–beverage combination is the product 20Γ—12=240.

As we have seen, applying the fundamental counting principle is rather straightforward. But can we apply it in situations where we have more than two independent events? Indeed we can. In fact, we can generalize it to situations where we have any number of events: if we have 𝑛 independent events 𝐴,𝐴,…,𝐴 which have 𝑝,𝑝,…,π‘οŠ§οŠ¨οŠ outcomes, respectively, the total number of distinct possible outcomes is π‘Γ—π‘Γ—β‹―Γ—π‘οŠ§οŠ¨οŠ. In these circumstances, applying the fundamental counting principle is as straightforward as in the case of two events as the next example will demonstrate.

Example 2: Using the Fundamental Counting Principle with Multiple Events

A skateboard shop stocks 10 types of board decks, 3 types of trucks, and 4 types of wheels. How many different skateboards can be constructed?

Answer

Using the fundamental counting principle, to find the total number of different skateboards we can construct, we can simply multiply together the number of choices for each part of the skateboard. Hence, the total number of distinct skateboards we can build is given by 10Γ—3Γ—4=120.

In the situation where we have multiple events, 𝐴,𝐴,…,𝐴, each with the same number of outcomes 𝑝, rather than writing

for the total number of distinct possible outcomes, we can simply write it as π‘οŠ.

Example 3: Fundamental Counting Principle with Multiple Events with the Same Number of Possible Outcomes

Jennifer is taking an online survey of nine yes/no questions. How many possible ways can Jennifer answer all the questions?

Answer

There are 9 questions each with two possible answers: yes and no. We might be tempted to think that the number of options is therefore 9Γ—2. However, this is incorrect. This would be the case if we had two events, one with two possible outcomes and the other with nine, whereas we have nine independent events, each with two possible answers. Hence, using the fundamental counting principle, we have a total of 2 distinct outcomes. Hence, the number of distinct ways that Jennifer can complete all the questions is 512.

In some situations we have a combination of events with the same number of outcomes and events with different numbers of outcomes. This is the case in the next example.

Example 4: Applying the Fundamental Counting Principle in Real-Life Situations

A code breaker is trying to find the value of an eight-digit number. The figure below shows the digits that he has already discovered. He has narrowed down his options for the digit represented by the letter 𝑐 to the following set of numbers: {5,6,4}. Given that he currently knows nothing about the other digits, how many possible numbers does he have left to try?

1796𝑐⋯⋯⋯

Answer

Since the code breaker knows the first four digits without any doubt, we only need to focus on the last four digits. The digit represented by 𝑐 can be one of 4, 5, and 6. Hence, there are three possible outcomes for the digit represented by 𝑐. As for the last three digits, they could be any number from 0 to 9. Hence, there are 10 possible outcomes for each of these. Therefore, applying the fundamental counting principle, the total number of numbers he has left to try is 3Γ—10=3,000.

Example 5: Fundamental Counting Principle with Compound Events

Suppose 10 fair coins are tossed at the same time these two spinners are spun. Using the fundamental counting principle, find the total number of possible outcomes.

Answer

We begin by considering the number of possible outcomes for each of the spinners. The first spinner has four colored regions; hence, there are four possible outcomes from it. As for the other spinner, there are eight distinct regions represented by the letters 𝐴 to 𝐻. Hence, there are eight possible outcomes for the second spinner. We now consider the ten coins. Each coin has two possible outcomes: heads and tails. Therefore, there are ten events each with two possible outcomes. Hence, using the fundamental counting principle, the total number of distinct outcomes is given by 2Γ—4Γ—8=32,768.

Key Points

  1. The fundamental counting principle enables us to find the total number of distinct outcomes of multiple independent events by finding the product of their individual possible outcomes.
  2. We can only apply the fundamental counting principle to independent events. If the outcome of one event changes the outcome of subsequent events, we must consider this effect when seeking to find the total number of possible outcomes.

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