Brian is going shopping for new blue jeans. He has many options from which to choose. For fit, he can choose skinny, boot cut, or flare. For wash, he can choose dark, faded, or distressed. For rise, he can choose low rise or regular. How many different options does Brian have?

In this concept, you will learn how to use the Counting Principle to calculate the total possible outcomes of a series of events.

### The Counting Principle

Tree diagrams provide you with a visual way of seeing all of the possible outcomes for a set of particular events. But it requires quite a bit of effort. What if there was a simpler way? You can use another principle to figure out the possible outcomes.

Let's look at an example:

Molly’s All Star Farm Breakfast features 3 choices of eggs–scrambled, fried, or omelet–plus a choice of bacon or sausage. You can use a tree diagram to find that there are 6 different choices, or outcomes, for the breakfast.

To look at this another way, you could look at the number of possible breakfast options in terms of outcomes.

For the first choice there are 3 different outcomes. For the second choice there are 2 different outcomes.

\begin{align*}3 \ \text{outcomes} \cdot 2 \ \text{outcomes} = 6 \ \text{outcomes}\end{align*}

The Counting Principle is a rule to figure out the total possible outcomes of a series of events. The Counting Principle for the number of choices or outcomes for two independent events, \begin{align*}A\end{align*} and \begin{align*}B\end{align*} taken together, is the product of the total number of outcomes for each event.

Total outcomes for \begin{align*}A\end{align*} and \begin{align*}B = (\text{number of outcomes for} \ A) \cdot (\text{number of outcomes for} \ B)\end{align*}

The Counting Principle will work for 2, 3 even 4 different events.

Here is another example:

For buying gum you have the following choices:

- 3 flavor choices–spearmint, peppermint, cinnamon
- 2 sugarless choices–sugarless or non-sugarless
- 2 bubble choices–bubble gum or regular

\begin{align*}3 \ \text{choices} \cdot 2 \ \text{choices} \cdot 2 \ \text{choices} = 12 \ \text{choices}\end{align*}

To check the answer, write out all of the possible options for gum:

\begin{align*}& \text{spear-sugarless-bubble} && \text{pepper-sugarless-bubble} && \text{cinnamon-sugarless-bubble}\\ & \text{spear-sugarless-regular} && \text{pepper-sugarless-regular} && \text{cinnamon-sugarless-}\\ & \text{spear-non-bubble} && \text{pepper-non-bubble} && \text{regular}\\ & \text{spear-non-regular} && \text{pepper-non-regular} && \text{cinnamon-non-bubble}\\ & && && \text{cinnamon-non-regular}\end{align*}

**Examples**

#### Example 1

Earlier, you were given a problem about Brian's blue jean shopping expedition.

He has a number of possible options which include fit: skinny, boot cut, or flare; wash: dark, faded, distressed; and rise: low or regular. How many possible choices of blue jeans does Brian have?

First, count the number of possibilities for each option:

Fit: 3

Wash: 3

Rise: 2

Next, use the Counting Principles method of calculating total outcomes:

\begin{align*}3 \ \text{choices} \cdot 3 \ \text{choices} \cdot 2 \ \text{choices} = 18 \ \text{choices}\end{align*}

Then, record your answer as total possible outcomes.

The answer is Brian has 18 choices of blue jeans from which to choose.

**Example 2**

You’re buying a sweater and have the following choices.

- Color choices–black, yellow, blue, red, green
- Material choices–wool, cotton, fleece
- Style choices–v-neck, crew, button-down, turtle

How many sweater possibilities do you have?

First, count the number of each different choice:

Color choices: 5

Material choices: 3

Style choices: 4

Next, use the Counting Principles method of calculating total outcomes:

\begin{align*}5 \ \text{choices} \cdot 3 \ \text{choices} \cdot 4 \ \text{choices} = 60 \ \text{choices}\end{align*}

Then, record your answer as total possible outcomes.

The answer is you have 60 choices of sweaters from which to choose.

#### Example 3

Omar is buying a skateboard. He has 5 different skateboard decks to choose from and 4 different wheel choices. How many different skateboard choices does Omar have?

First, count the number of each different choice:

Decks: 5

Wheels: 4

Next, use the Counting Principles method of calculating total outcomes:

\begin{align*}5 \ \text{choices} \cdot 4 \ \text{choices} = 20 \ \text{choices}\end{align*}

Then, record your answer as total possible outcomes.

The answer is Omar has 20 different skateboard choices.

#### Example 4

Ice Stone ice cream shop has 3 different sundae sizes: baby, large, and grand. You can choose from 6 different ice cream flavors and add 4 different toppings. How many sundae choices are there?

First, count the number of each different choice:

Size: 3

Flavors: 6

Toppings: 4

Next, use the Counting Principles method of calculating total outcomes:

\begin{align*}3 \ \text{choices} \cdot 6 \ \text{choices} \cdot4 \ \text{choices} = 72 \ \text{choices}\end{align*}

Then, record your answer as total possible outcomes.

The answer is you have 72 choices of sundaes.

#### Example 5

Gina tosses a number cube 2 times. How many different outcomes are there?

First, count the number of each different choice:

Number choices: 6

Number of rolls: 2

Next, use the Counting Principles method of calculating total outcomes:

\begin{align*}6 \ \text{choices} \cdot 2 \ \text{rolls} = 12 \ \text{outcomes}\end{align*}

Then, record your answer as total possible outcomes.

The answer is there are 12 outcomes.

### Review

Use the Counting Principle to solve each problem.

- The Cubs have 3 games left to play this year. How many different outcomes can there be for the three games?
- Svetlana tosses a coin 4 times in a row. How many outcomes are there for the 4 tosses?
- For a new tennis racquet, Danny can choose from 8 different brands, 3 different head sizes, and 4 different grip sizes. How many different racquet choices does Danny have?
- Gina tosses a number cube 3 times. How many different outcomes are possible?
- Gina tosses a number cube. Buster flips a coin. How many different outcomes are possible for the two events?
- Buster flips a coin. Daoud chooses a card from a deck of 52 cards. How many different outcomes are possible for the two events?
- Rex spins a spinner that has red, blue, and yellow sections two times. How many different outcomes are possible?
- Daoud chooses a card from a deck of 52 cards, replaces the card in the deck, then chooses a second card. How many different outcomes are possible?
- Patsy’s Pizza features 3 different pizza types, 14 different toppings, and 2 different sizes. How many different pizzas can you order?
- Spud has a 2-letter password for his computer using letters only. If there are 26 different letters in the alphabet, how many different passwords are possible?
- Doreen has a 3-digit password for her computer using digits only. If there are 10 different digits (including zero), how many different passwords are possible?
- Sebastian has a 3-letter password for his computer using vowels (A, E, O, I, U) only. How many different passwords are possible?
- If Sebastian had a two - letter password, how many different passwords are possible?
- If Sebastian had a four - letter password, how many different passwords are possible?
- If Sebastian had a five - letter password, how many different passwords are possible?

### Review (Answers)

To see the Review answers, open this PDF file and look for section 12.10.