# 1.2: Fundamental Counting Principle

**At Grade**Created by: Bruce DeWItt

### Learning Objectives

- Apply the Fundamental Counting Principle to determine the number of outcomes
- Create tree diagrams to represent outcomes for a series of events

The **Fundamental Counting Principle** states that if you wish to find the number of outcomes for a given situation, simply multiply the number of outcomes for each individual event. In Example 2 in section 1.1, the child had two different choices of drink and three different choices of meat. If we multiply 2 times 3, we get 6 which is the total number of outcomes possible. The Fundamental Counting Principle expands to any number of events. For example, suppose it turned out that the child also wanted to order eggs and had a choice between scrambled and sunny-side up. The fundamental counting principle states that there are \begin{align*}2\times3\times2\end{align*}

There are other ways to visually see what is happening here. Let's use a **tree diagram**.

#### Example 1

Build a tree diagram that shows the different outcomes for what the child might order for breakfast.

#### Solution

The first set of branches of the tree diagram will represent the type of drink, the second set of branches will represent the type of meat, and the third set of branches will represent the type of egg. A diagram of what this will look like is shown on the top of the next page.

We have labeled the ends of two of the branches in the figure above to show what each branch means. For example, one of the labeled branches shows that the child might have ordered milk, bacon, and scrambled eggs.

The Fundamental Counting Principle is critically important especially when considering complex tree diagrams. Our tree diagram above has many branches and it tracks a great deal of material. It ultimately shows us the 12 different possible breakfast orders, but it takes a large amount of organization to successfully complete. Multiplying 2 by 3 by 2 is a much quicker way to find out the total number of possible outcomes.

#### Example 2

A couple is planning to have 3 children. Consider the different results that might occur in terms of gender. For example one outcome might be Boy, Boy, Girl (BBG).

a) Using the Fundamental Counting Principle, calculate the number of different outcomes for the children in this family.

b) Build a tree diagram that shows the different orders of children the couple might have.

c) Construct the sample space that shows all the different orders of children the couple might have.

#### Solution

a) There are 2 choices for the first child, 2 for the second, and 2 for the third. Therefore, there are \begin{align*}2\times2\times2=8\end{align*}

2×2×2=8 outcomes for the gender order of the 3 children.

b)

c) In order to be organized, the list will be alphabetized. BBB, BBG, BGB, BGG, GBB, GBG, GGB, GGG There are a total of 8 outcomes.

There are many other ways to apply the Fundamental Counting Principle. A standard deck of cards has 52 cards as shown below. If you are dealt just one card, there are 52 different outcomes.

#### Example 3

Suppose you are dealt two cards from a standard deck of 52 cards. How many different outcomes are possible?

#### Solution

We could certainly try drawing a tree diagram but that could get very large quite quickly. The first split alone would have 52 branches on it. On the other hand, if we use the Fundamental Counting Principle, we can simply calculate how many different ways we could be dealt 2 cards from a standard deck. There would be 52 choices for the 1st card and 51 choices for the 2nd card. (Once the first card is dealt, the deck only has 51 cards left in it.) There are \begin{align*}52\times51=2652\end{align*}

52×51=2652 ways that we could be dealt two cards from the deck.

#### Example 4

How many different 7-digit phone numbers are possible if no phone number may begin with a zero?

#### Solution

There are a total of 10 digits available {0, 1, 2, ... 7, 8, 9}. We can't use zero for the first digit so there are only 9 choices for the 1st digit. After that, there are 10 digits available for each of the remaining six digits. This gives us \begin{align*}9\times10\times10\times10\times10\times10\times10=9\times10^6=9,000,000\end{align*}

9×10×10×10×10×10×10=9×106=9,000,000 ways to come up with a 7 digit phone number.

#### Example 5

A teenager is given 5 different jobs that they must do before they may go out to a movie with friends. The jobs are washing the car, starting a load of laundry, vacuuming the family room, taking out the garbage, and putting away the dishes. In how many different orders could the teenager complete these jobs?

#### Solution

There are five choices the teenager could pick for the first job. Once that job is finished, there are only 4 jobs remaining. Once the 2nd job is completed, there are only 3 choices for the 3rd job. Once the 3rd job is finished, there are only 2 choices for the 4th job and finally there will only be one choice left for the 5th job. There are \begin{align*}5\times4\times3\times2\times1=120\end{align*}

5×4×3×2×1=120 different orders that these jobs could be completed. Note that there is a quick way to do this ordered multiplication usingfactorials. \begin{align*}5\times4\times3\times2\times1=5!\end{align*}5×4×3×2×1=5! The 5! is read "Five Factorial". Be sure to locate the factorial key on your calculator.

### Problem Set 1.2

#### Exercises

1) A woman has three skirts, five shirts, and four hats. How many different outfits can she wear if she picks one skirt, one shirt, and one hat for her outfit?

2) How many different five-digit ZIP codes are possible if the digits can be repeated?

3) How many different five-digit ZIP codes are possible if the digits cannot be repeated?

4) In how many ways can a baseball manager arrange a batting order of nine players?

5) A store manager wishes to display six different brands of laundry soap by lining them up in a row on a shelf. In how many ways can this be done?

6) There are 8 different statistics books, 6 different geometry books, and 3 different trigonometry books being considered for next year. In how many ways can a textbook committee select one of each book?

7) At a film festival, there are eight different films that will be shown. In how many different orders can these films be shown?

8) The call letters of a radio station must have four letters. The first letter must be a K or a W. How many different call letter combinations are possible if letters may not be repeated?

9) The call letters of a radio station must have four letters. The first letter must be a K or a W. How many different call letter combinations are possible if letters may be repeated?

10) How many different four-digit ID tags can be made if repeats are allowed?

11) How many different four-digit ID tags can be made if it must start with a 7 and no repeats are allowed?

12) In how many different ways can the Harry Potter series of books (7 books total) be arranged in a row on a shelf?

13) In how many different ways can a manager select a pitcher - catcher combination if the manager has 5 pitchers and 2 catchers to choose from?

14) A coin is tossed 8 times. How many different outcomes are there for this series of 8 flips?

15) Six different colored tiles are available to make a pattern in a row of floor tile. How many possible different 4-color patterns are possible if no colors may be repeated?

16) Six different colors of tile are available to make a pattern in a row of floor tile. Many tiles of each color are available. How many 4-color patterns can be made if colors may be repeated?

17) Four cards are dealt from a standard deck of 52 cards. In how many different orders of suit could the cards be dealt? For example, one order is Club, Heart, Club, Diamond.

18) A pizza restaurant offers 6 different toppings for their pizzas. How many different pizzas are possible?

19) Use a tree diagram to find all possible outcomes for the result of a series of coin flips if the coin is flipped two times. Write a list of the possible results when complete.

20) The Super-Cool Ice Cream Shoppe sells sundaes, cones, or ice cream bars. You will pick either butterscotch or chocolate and you may choose to have it with nuts or without nuts.

a) Draw a tree diagram to illustrate the different types of ice cream treats that you could order.

b) How could you find the number of outcomes using the Fundamental Counting Principle?

c) How many different outcomes are possible?

21) A quiz has four true/false questions on it. Use a tree diagram to show all the different possible answer keys.

22) A box contains a $1 bill, a $5 bill, and a $10 bill. Two bills are selected one after the other without replacing the first bill. Draw a tree diagram to show all possible amounts of money that may be drawn.

23) The Eagles and Hawks play each other in a hockey tournament. The first team to win two games is the champion. Use a tree diagram to show all different possible outcomes for the tournament.

#### Review Exercises

24) Consider a situation in which a baseball manager must decide which one of 4 players will pitch (P1, P2, P3, or P4) and which one of 2 players will catch (C1 or C2).

a) What is the sample space for this situation?

b) How many outcomes are possible?

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