# 12.10: Basic Counting Rules

**At Grade**Created by: CK-12

**Practice**Basic Counting Rules

Remember Alicia’s outfit? Well, when Alicia was working on figuring out the number of possible outfits that she could create, we created a tree diagram. A tree diagram is very useful because it provides us with a visual display of the data.

From the tree diagram, we could count all of the possible outcomes. There are 12 possible outfits for Alicia to choose from.

What about if we didn’t want to draw a tree diagram? Is there another way that we could have thought about figuring out the number of outfits possible?

**This Concept is all about The Counting Principle. The Counting Principle makes figuring outcomes possible because of a mathematical method not a tree diagram. Use this lesson to learn about The Counting Principle and we will apply it to Alicia’s outfits at the end of the Concept.**

### Guidance

Tree diagrams provide you with a visual way of seeing all of the possible outcomes for a set of particular events.

**What if there was a simpler way?**

Sometimes, you don’t want to have to draw a tree diagram to figure out all of the possible outcomes for a series of events. When this happens, we can use another principle to figure out the possible outcomes.

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.

**What if we wanted to look at this in another way?**

We 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*}

**Yes they are. That is because this is a new principle to figure out the total possible outcomes of a series of events. We call in the** *Counting Principle.*

**Counting Principle:** The number of choices or outcomes for two independent events, \begin{align*}A\end{align*}

Total outcomes for \begin{align*}A\end{align*}

Now that you know what the Counting Principle is, you can practice using it. The Counting Principle will work for 2, 3 even 4 different events. Just follow the procedure of using multiplication to find the number of possible outcomes.

Now let’s apply the Counting Principle to a scenario.

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

**To find the number of gum choices you have you could make a tree diagram, or you could simply use the Counting Principle:**

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

To check our answer, we can 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*}

**You can see that the Counting Principle worked out fine for our solution.**

Now it's your turn to practice. Use the Counting Principle to count outcomes.

#### Example A

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?

**Solution: 20 skateboard choices**

#### Example B

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?

**Solution: 72 sundae choices**

#### Example C

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

**Solution: 12 outcomes**

**Here is the original problem once again. Reread it and then we will look at applying The Counting Principle to this problem.**

Remember Alicia’s outfit? Well, when Alicia was working on figuring out the number of possible outfits that she could create, we created a tree diagram. A tree diagram is very useful because it provides us with a visual display of the data.

From the tree diagram, we could count all of the possible outcomes. There are 12 possible outfits for Alicia to choose from.

What about if we didn’t want to draw a tree diagram? Is there another way that we could have thought about figuring out the number of outfits possible?

**Think back to the lesson about The Counting Principle.**

**Total Outcomes = (Number of outcomes)(Number of outcomes)**

**With Alicia’s problem, there are three possible numbers of outcomes. We have the shirts that she selected as options, there are two of them. We have the skirts that she selected as options, there are three of those. We have the shoes that she selected as options, there are two of those.**

\begin{align*}3 \times 2 \times 2 = 12 \ \text{possible outfits}\end{align*}

**You can see that we ended up with the same answer as we did with the tree diagram. The Counting Principle definitely works.**

### Vocabulary

Here are the vocabulary words found in this Concept.

- Probability
- the ratio of favorable outcomes to total possible outcomes.

- The Counting Principle
- the product of the outcomes of a series of events gives the total number of outcomes.

### Guided Practice

Here is one for you to try on your own.

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

- 5 different color choices–black, yellow, blue, red, green
- 3 different material choices–wool, cotton, fleece
- 4 different style choices–v-neck, crew, button-down, turtle

**To find the number of sweater choices you have you could make a tree diagram, or you could simply use the Counting Principle:**

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

### Video Review

Here is a video for review.

- This is a James Sousa video on probability.

### Practice

Directions: Use the Counting Principle to solve each problem.

1. The Cubs have 3 games left to play this year. How many different outcomes can there be for the three games?

2. Svetlana tosses a coin 4 times in a row. How many outcomes are there for the 4 tosses?

3. 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?

4. Gina tosses a number cube 3 times. How many different outcomes are possible?

5. Gina tosses a number cube. Buster flips a coin. How many different outcomes are possible for the two events?

6. Buster flips a coin. Daoud chooses a card from a deck of 52 cards. How many different outcomes are possible for the two events?

7. Rex spins a spinner that has red, blue, and yellow sections two times. How many different outcomes are possible?

8. 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?

9. Patsy’s Pizza features 3 different pizza types, 14 different toppings, and 2 different sizes. How many different pizzas can you order?

10. 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?

11. 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?

12. Sebastian has a 3-letter password for his computer using vowels (A, E, O, I, U) only. How many different passwords are possible?

13. If Sebastian had a two - letter password, how many different passwords are possible?

14. If Sebastian had a four - letter password, how many different passwords are possible?

15. If Sebastian had a five - letter password, how many different passwords are possible?

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Term | Definition |
---|---|

factorial |
The factorial of a whole number is the product of the positive integers from 1 to . The symbol "!" denotes factorial. . |

inferential statistics |
With inferential statistics, your goal is use the data in a sample to draw conclusions about a larger population. |

multiplicative rule of counting |
The Multiplicative Rule of Counting states that if there are n possible outcomes for event A and m possible outcomes for event B, then there are a total of nm possible outcomes for the series of events A followed by B. |

Probability |
Probability is the chance that something will happen. It can be written as a fraction, decimal or percent. |

The Counting Principle |
In probability, the counting principle states that the number of outcomes for two independent events taken together is the product of the total number of outcomes for each individual event. |

### Image Attributions

Here you'll recognize and use The Counting Principle to find all possible outcomes.