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Fundamental Counting Principle

- Calculus
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Suppose you are about to order a pizza. You can choose between 3 different types of crusts, 8 different toppings, and 3 different types of cheeses. Given these conditions, how many kinds of pizzas are possible for your order? Well, you can try listing down all the individual possibilities and summing them up. That is clearly a very tedious and inefficient way of going about it. A much better alternative is to use the **fundamental counting principle.**

The **fundamental counting principle** is used to determine the number of outcomes possible in a given situation, as related to probability.

The fundamental counting principle states that:

If there are m possible outcomes for event M and n possible outcomes for event N, then the number of possible outcomes where event M is followed by event N is m × n.

When speaking about probability, what exactly is an event? An **event** is an outcome or a set of multiple outcomes, which can be assigned a probability of occurrence in a statistical experiment. For example, in the instance of tossing a die, the toss itself is the event, while the possible resulting numbers (1 through 6) make up the set of possible outcomes. There are two types of events which we will discuss in relation to the fundamental counting principle: independent and dependent events.

**Independent** **events** do not affect the probability of occurrence of other events, and their likelihoods of occurrence are not affected by other events either. For example, the choices of digits in a phone number are independent events since each digit chosen does not affect the choices for the others.

On the other hand, the probability of occurrence for** dependent events** is influenced by and dependent on the outcome of another event. Conversely, dependent events may also affect other events' likelihoods. For example, the winners of the second round of a knockout tennis tournament depend on the outcomes of the first round matches. Hence, the outcome of the second round is dependent on the outcome of the first round.

The distinction between these two types of events is important when using the fundamental counting principle because whether an event is independent or dependent influences its number of possible outcomes. If an event is dependent, its number of outcomes may be limited.

Let us look at some examples to better understand the fundamental counting principle and how to apply its formula. We will explore how it is used for both independent and dependent events.

A sandwich cart offers customers a choice of hamburger, chicken, or fish on either a plain or a sesame seed bun. How many different combinations of meat and bun are possible?

**Solution:**

Here, we can have 3 different types of meat and 2 different types of buns. As the situation calls for a series or combination of selections to be made, the fundamental counting principle can be applied here. The selection of meat type is not affected by the selection of bun type, which makes these two selections **independent events**. To arrive at the total number of possible combinations, first we must ask ourselves how many events there are in this situation and how many outcomes are associated with each event.

Event 1: A meat type is selected.

- 3 outcomes: This selection could result in any of the three choices of hamburger, chicken, or fish.

Event 2: A bun is selected.

- 2 outcomes: This selection could result in any of the two choices of a plain bun or a sesame bun.

According to the fundamental counting principle, this means there are 3 × 2 = 6 possible combinations (outcomes).

The fundamental counting principle can be used for cases with more than two events. For example, if there are 4 events E1, E2, E3, and E4 with respective O1, O2, O3, and O4 possible outcomes, then the total number of possibilities of all four events taken together would be calculated as O1 × O2 × O3 × O4. Let's look at an example problem which calculates the possible outcomes of three events taken together.

You are about to order a pizza. You can choose between 3 different types of crusts, 8 different toppings, and 3 different types of cheeses. How many kinds of pizzas can you order?

**Solution:**

As the selection of crusts, toppings, and cheese do not affect one another, we know that these three selection events are independent.

Event 1: A crust type is selected.

- 3 outcomes: This selection could result in any of the three choices. ( )

Event 2: A topping is selected.

- 8 outcomes: This selection could result in any of the 8 choices of toppings. ( )

Event 3: A cheese is selected.

- 3 outcomes: This selection could result in any of the 3 choices of cheese. ( )

Hence, by the f**undamental counting principle, **the total amount of pizzas which can be made with the above choices are: .

The fundamental counting principle can also be demonstrated using a tree diagram, which helps us to consider the possible outcomes of events from a visual perspective. Let's revisit our first example problem and create a tree diagram to analyze it visually. Suppose H represents hamburger, C is for chicken, and F is for fish:

For each of these three choices of meat, we have two subsequent "branches" of the tree diagram which show the possible outcomes of the next selection event, with the options of a plain bun (P) and a sesame seed bun (S). The lowermost nodes of the tree (also known as leaves of the tree) give each possible outcome of the experiment as a whole, of which there are 6: HP, HS, CP, CS, FP, and FS.

The examples we have seen so far have involved independent events. The fundamental counting principle can be applied to dependent events as well, however. Let's look at an example which deals with dependent events.

John's school offers 8 periods each day, and he must choose 4 subjects in total for the school year. He has to create a schedule with 1 class of each subject every day. Assuming all subjects are available during each period, how many possible schedules can John choose from?

**Solution:**

When John schedules a given class for a given period, he cannot schedule that class for any other period. Therefore, the choices are dependent events.

Event 1: John schedules the 1st subject.

- 8 outcomes: He can schedule the 1st subject in 8 different slots (periods). ()

Event 2: John schedules the 2nd subject.

- 7 outcomes: Now that he scheduled the 1st subject, he has 7 options for the 2nd subject.

Event 3: John schedules the 3rd subject.

- 6 outcomes: Now that the 1st and 2nd subjects are scheduled, 6 options remain for the 3rd. ()

Event 4: John schedules the 4th subject.

- 5 outcomes: Now that 1st, 2nd, and 3rd subjects are scheduled, 5 options remain for the 4th. ()

Thus, according to the fundamental counting principle, the total number of possible schedules is:

8 × 7 × 6 × 5 = 1,680.

Hence, John has 1,680 possible choices to schedule his classes.

While there are multiple methods for calculating permutations and combinations, one of the options is to use the fundamental counting principle.

First, let us clarify the difference between permutations and combinations. Permutations and combinations both deal with the topic of selecting a certain number of objects from a given set of objects. So, suppose there is a trial for a football team in which 100 people try out, but we have to choose 11 out of those 100. Assuming the **order **in which those 11 players are selected **does not matter**, it is a **combination**. If the **order** in which we select those players **does matter,** it is known as a permutation.

Now, let's use the fundamental counting principle to calculate the possible outcomes of a combination problem.

How many numbers are there between 100 and 999 whose middle digit is 4?

**Solution:**

Given that the middle digit is fixed, the 2 events here are the selection of the leftmost and rightmost digits. The leftmost digit can be selected in 9 possible ways (1 to 9), and the rightmost digit can be selected in 10 possible ways (0 to 9). The events are independent of each other.Thus, according to the fundamental counting principle, the total possible numbers is:9 × 10 = 90 numbers

- The
**fundamental counting principle**is used to determine the number of outcomes possible in a given situation, as related to probability. - If there are m ways for event M to occur and n ways for event N to occur, then event M followed by event N can occur in m × n ways.
**Independent****events**are those events whose probability of occurrence is not dependent on any other event.**Dependent events**are those events where the probability of occurrence of one event depends on the outcome of another event.- The fundamental counting principle can be extrapolated to cases with multiple events.
- The fundamental counting principle can be applied to both dependent and independent events.

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