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12.7: Counting Events

Difficulty Level: At Grade Created by: CK-12
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Let's Think About It

Jessica is excited to attend her Halloween middle school dance.  There is no dress code, so her optionns for the outfit she will wear is wide open.  She has limited her choices to the following:

Tops:  a lacy blouse or a graphic t-shirt

Bottoms:  skinny jeans or an A-line skirt

Shoes:  platform sandals or Vans tennis shoes

How many different outfits can Jessica create given these options?

In this concept, you will learn how to calculate the number of outcomes that exist when given different possibilities.

Guidance

A good way to find the total number of outcomes for events is to make a tree diagram. A tree diagram is a branching diagram that shows all possible outcomes for an event. To make a tree diagram, split the different events into either-or choices. You can list the choices in any order.

Let's look at an example.

Nadia’s soccer team has 2 games to play this weekend. How many outcomes are there for Nadia’s team?Here is a tree diagram for game 1 and game 2.  Below that is a tree diagram illustrating the possible outcomes if Nadia were to play 3 games.

As you can see, there are four different outcomes for the two games:

win-win           win-lose              lose-win                 lose-lose

If Nadia played 3 games, there would be 8 different outcomes:

win-win-win                 win-lose-win                lose-win-win                   lose-lose-win

win-win-lose                win-lose-lose               lose-win-lose                  lose-lose-lose

A tree diagram is a great way to visually see all of the options possible. It can also help you to organize the outcomes so that you don’t miss any possibilities.

To remodel her kitchen, Gretchen has the following choices: Floor: tile or wood; Counter: Granite or formica; Sink: white, steel, stone. How many different choices can Gretchen make?

First, list the choices:

Choice 1 Floor: tile, wood Choice 2 Counter: granite, formica Choice 3 Sink: white, steel, stone

Next, start a tree diagram by listing any of the optioins for Choice 1, branching off to options for Choice 2, and finally Choice 3. Make sure Choice 2 repeats for each branch of Choice 1, and that Choice 3 repeats for each branch of Choice 2.

Some spaces are left for you to fill in. Double click to check your answers.

Then fill in the outcomes:

The answer is that there are 12 possible outcomes for the kitchen design.

While tree diagrams provide a visual layout of choice options and outcomes, they can take time to create.  Another quicker way to calculate the number of possible outcomes when provided different categories of choices is to multiply together the number of choice options for each category.

For example, Kelly has chocolate or vanilla ice cream, three choices of toppings and four sauces.  How many possible outcomes does she have for creating her ice cream sundae?

 \begin{align*}2\ (ice cream)\ x\ 3\ (toppings)\ x\ 4\ (sauces)\ = 24\ possible\ outcomes\end{align*}

Guided Practice

A local pizza place offers a pizza with several options. On a basic pizza, you can choose one of each option. The options are a sourdough or whole wheat crust, mozzarella or feta cheese, and one of the following toppings: black olives, green olives, green pepper, garlic, pepperoni, ground beef, mushrooms, or onions, .

Based on these possibilities, how many possible outcomes are there?

First, list all of the choices:

Crust: sourdough or whole wheat

Cheese:  mozzarella or feta

Toppings:  black olives, green olives, green pepper, garlic, pepperoni, ground beef, mushrooms, or onions

Next, count the number of options for each category:

Crust: 2

Cheese:  2

Toppings:  8

Then, multiply these numbers together to get the number of possible outcomes:

\begin{align*}2 \times 2 \times 8 = 32\end{align*}

The answer is there are \begin{align*}32\end{align*} different pizza options.

Examples

Example 1

Jeff has five different pairs of socks and three pairs of shoes. How many possible combinations are there?

First, list all of the choices:

Socks: 5 different pairs

Shoes:  3 different pairs

Next, list the number of options for each category:

Socks: 5

Shoes: 3

Then, multiply these numbers together to get the number of possible outcomes:

\begin{align*}5 \times 3 = 15\end{align*}

The answer is there are 15 possible combinations of shoes and socks.

Example 2

Jessie has three sweaters, two turtlenecks and three jackets. How many possible combinations are there?

First, list all of the choices: 

3 sweaters

2 turtlenecks

3 jackets

Next, list the number of options for each category:

Sweaters:  3

Turtlenecks:  2

Jackets:  3

Then, multiply these numbers together to get the number of possible outcomes:

\begin{align*}3 \times 2 \times 3 = 18\end{align*}

The answer is Jessie can make 18 different combinations.

Follow Up


Remember Jessica and her outfit for the middle school Halloween dance?  

She had limited her choices to a lacy blouse or a graphic t-shirt, skinny jeans or an A-line skirt, and platform sandals or Vans tennis shoes.  How many combinations can Jessica make:

First, list all of the choices and options for each:

Tops:  lacy blouse or graphic t-shirt

Bottoms:  skinny jeans or A-line skirt

Shoes:  platform sandals or Vans tennis shoes

Next, list the number of options for each category:

Tops:  2

Bottoms:  2

Shoes:  2

Then, multiple these numbers together to get the number of possible outcomes:

2 x 2 x 2 = 8

The answer is Jessica has 8 different outfit combinations from which to choose for the dance.

Video Review

Explore More

Use tree diagrams to figure out the total number of possible outcomes for each situation.

1. The Triplex Theater has 3 different movies tonight: Bucket of Fun, Bozo the Great, and Pickle Man. Each movie has an early and late show. How many different movie choices are there?

2. Raccoon Stadium offers the following seating plans for football games:

  • lower deck, middle loge, or upper bleachers
  • center, side, end-zone

How many different kinds of seats can you buy?

3. Cell-Gel cell phone company offers the following choices:

  • Free internet plan or Pay internet plan
  • 1200, 2000, or 3000 minutes
  • Premium or standard phone

How many different kinds of plans can you get?

4. Jen’s soccer team is playing 4 games next week. How many different outcomes are there for the four games?

5. The e-Box laptop computer offers the following options.

  • Screen: small, medium, or large
  • Memory: standard 1 GB, extra 2 GB
  • Colors: pearl, blue, black

How many different combinations are there for options?

6. Jesse has four pairs of pants and three shirts. How many different outfits can he make?

7. If Jesse adds three sweaters, how many different outfits can he make now?

8. Kyle has selected three different paint colors and two trim colors for his bedroom. How many different color combinations can he create?

9. There are four teams playing in a tournament. Each team will play two sports. How many different combinations are there for teams and sports?

10. Candice can choose between the following electives.

  • German
  • Gymnastics
  • Chorus
  • French

If she can choose two electives, how many different options are there of choices?

11. If we add Track to her options, how many two elective options are there?

12. Miles is making a salad. He wants to put three things on it. Here are his choices.

Tomatoes

Pickles

Onions

Cucumbers

Peppers

How many different salads can he create given these options?

13. If Miles adds cheese, how many different salads can he create given these options?

14. Muffins can have two additions to them. Here are the choices: blueberries, raspberries, nuts, cherries, strawberries. How many muffin combinations can there be?

15. If we add apples and cranberries, how many combinations can there be now?

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Date Created:
Aug 10, 2015
Last Modified:
Mar 23, 2016
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