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Introduction

The Ferris Wheel

Maggie, Sarah and Julie are excited to go on the Ferris wheel. There isn’t any line, and so the friends decide to ride the Ferris wheel multiple times in a row. It is great! The Ferris wheel stops at the top and they can see all across the entire park. Sarah spots their teacher Mrs. Hawk and gives a huge wave. The others join in.

Each seat can only hold two people so the friends take turns sitting with each other. They keep riding the Ferris wheel until everyone has had a chance to sit with everyone else. After the last ride, they get off the ride, a little dizzy, but very happy!

“Wow that was some time!” Maggie says excitedly.

“Yes, but my head is still spinning,” Julie declares.

As they walk away, Chris comes over. When he asks where they have been, they tell him that they have been riding the Ferris wheel.

“How many times did you ride it?” Chris asks.

All three of the friends look at each other. They aren’t sure. It was so exciting to keep riding that they lost count.

“I know we can figure this out mathematically,” Maggie says to the others as she starts to count on her fingers.

Do you know how many times they rode the Ferris wheel? If each friend rode with each other once, how many times did they ride in all? You can use a few different methods to figure out this outcome. In this lesson, you will learn all about finding outcomes. Pay attention so that you can figure this problem out in the end.

What You Will Learn

By the end of this lesson you will be able to demonstrate the following skills:

  • Use tree diagrams to list all possible outcomes.
  • Find all possible combinations.
  • Find all possible permutations.
  • Describe real-world situations involving combinations or permutations.

Teaching Time

I. Use Tree Diagrams to List all Possible Outcomes

When thinking about probability, you think about the chances or the likelihood that an event is going to occur. Calculating probability through a ratio is one way of looking at probability. We can also think about chances or probability through calculating outcomes.

What is an outcome?

An outcome is an end result. When you have multiple options you can calculate an outcome or figure out how many possible outcomes there are. We do this all the time in life and we don’t even realize that we are doing it. Anytime you are trying to organize something with many different pieces or components, you are figuring outcomes.

How can we figure out an outcome?

There are a couple of different ways to do this, and you are going to learn about them in this lesson. The first one that we are going to work with is a tree diagram.

What is a tree diagram?

A tree diagram is a visual way of listing outcomes. You look at the choices for the outcome and the variables that go with each outcome.

Alright, let’s slow down. First, let’s look at an example and that will help make tree diagrams a lot clearer.

Example

Jessica has four different favorite types of ice cream. She loves vanilla crunch, black raspberry, chocolate chip and lemonade. She also loves two different types of cones, a plain cone and a sugar cone. Given these flavors and cones choices, how many different single scoop ice cream cones can Jessica create?

To solve this problem, we are going to create a tree diagram.

First, we list the choices of ice cream.

Vanilla Crunch

Black Raspberry

Chocolate Chip

Lemonade

Next we add in the two cone types. Each flavor has two possible cone types that it could go on. This is where the tree diagram part comes in.

Here we have four different flavors, and two types of cones, which means we have 8 possible ice cream cone options.

Did you notice any patterns here?

If you pay close attention, you can see that the number of choices multiplied by the number of variables gave us the total number of outcomes.

4 \times 2 = 8

This is called the Fundamental Counting Principle and it can be very useful if you don’t want to draw an elaborate diagram to figure out your options!!

Practice finding outcomes. You may draw a tree diagram or use the Fundamental Counting Principle to answer each question.

  1. Sarah has three pairs of pants and four shirts. How many different outfits can she create with these choices?
  2. Travis has four different pairs of striped socks and two pairs of sneakers, one red and one blue. How many different shoe/sock combinations can Travis create?
  3. If there are 33 ice cream flavors and two types of cones, how many different single scoop ice cream cones can you create?

Take a few minutes to compare your work with a friend’s work. Are there any differences in your approaches? How about in your answers?

II. Find All Possible Combinations

When you have a combination, order does not matter. The ice cream cones were a good example. It did not matter what the order was of the flavors or the cones. We just wanted to know how many different possible cones could be created.

We can find all of the possible combinations when working with examples.

How do we do that?

We work on figuring out combinations by listing out all of the possible options. Then we eliminate any duplicates and the number of outcomes left is our answer. Let’s look at an example.

Example

Seth, Keith, Derek and Justin want to go on the bumper cars. They can only ride in pairs. How many different paired combinations are possible given these parameters?

To start, we list out all possible options beginning with Seth. Seth can ride with Keith Derek or Justin. Keith can ride with Seth, Derek or Justin. Derek can ride with Seth, Justin or Keith. Justin can ride with Seth, Derek or Keith.

Here are the possible combinations.

&\text{SK} && \text{KS} && \text{DS} && \text{JS}\\&\text{SD} && \text{KD} && \text{DK} && \text{JK}\\&\text{SJ} && \text{KJ} && \text{DJ} && \text{JD}

Next we cross out any duplicates.

&\text{SK} && \text{\bcancel{KS}} && \text{\bcancel{DS}} && \text{\cancel{JS}}\\&\text{SD} && \text{KD} && \text{\cancel{DK}} && \text{\cancel{JK}}\\	&\text{SJ} && \text{KJ} && \text{DJ} && \text{\cancel{JD}}

There are six different pair combinations.

What if order had made a difference? What if we had wanted to count each person if they sat in a different seat? What would have happened then?

That is where our next way of figuring outcomes comes in. It is called permutations.

III. Find All Possible Permutations

A permutation is a combination where order makes a difference. In the last section, we didn’t care about order. We just cared about the pairings.

What if we had cared about the order?

If we had cared about order, then SK and KS would be two different things.

We would have counted ALL of the possible combinations and they would be included in our permutation because order matters.

Let’s look at the permutations from the last problem.

&\text{SK} && \text{KS} && \text{DS} && \text{JS}\\&\text{SD} && \text{KD} && \text{DK} && \text{JK}\\	&\text{SJ} && \text{KJ} && \text{DJ} && \text{JD}

Here we have 12 possible outcomes for this permutation.

Is there any easier way to figure this out besides writing out all of the possibilities?

Yes there is. In fact, there is a way to do this using specific notation.

First, we had four boys in pairs. Four taken two at a time, here is our permutation.

P(4,2)

This tells us that we have four options taken two at a time.

We figure out the permutation by counting down from four two numbers and we multiply them.

4 \cdot 3

Notice that we muliply the last two digits in the count up to four. There are two numbers to multiply because the boys were arranged two at a time. Next we multiply.

4 \times 3 = 12

There are 12 possible combinations. That is the same answer that we found by writing things all out.

Let’s look at another example.

Example

How many ways can you arrange five swimmers in groups of three?

This time we have groups of 3, so we multiply together the last 3 numbers in the count up to our number of items. Here is the permutation of 5 taken three at a time.

P(5, 3) = 5 \times 4 \times 3

There are 60 possible combinations.

Practice figuring out the following permutations.

  1. P(9, 2)
  2. P(4, 3)
  3. P(5, 2)

Take a few minutes to check your work with a peer.

Take a few notes on the difference between a permutation and a combination!

Real Life Example Completed

The Ferris Wheel

Here is the original problem once again. Reread it and underline any important information.

Maggie, Sarah and Julie are excited to go on the Ferris wheel. There isn’t any line, so the friends decide to ride the Ferris wheel multiple times in a row. It is great! The Ferris wheel stops at the top and they can see all across the entire park. Sarah spots their teacher Mrs. Hawk and gives a huge wave. The others join in.

Each seat can only hold two people, so the friends take turns sitting with each other. They keep riding the Ferris wheel until everyone has had a chance to sit with everyone else. After the last ride, they get off the ride, a little dizzy, but very happy!

“Wow that was some time!” Maggie says excitedly.

“Yes, but my head is still spinning,” Julie declares.

As they walk away, Chris comes over. When he asks where they have been, they tell him that they have been riding the Ferris wheel.

“How many times did you ride it?” Chris asks.

All three of the friends look at each other. They aren’t sure. It was so exciting to keep riding that they lost count.

“I know we can figure this out mathematically,” Maggie says to the others as she starts to count on her fingers.

Thinking about tree diagrams, combinations and permutations, how can Maggie figure this out mathematically?

We could use a tree diagram to figure this out. We could also write out all of the combinations.

When order matters, we know that we are going to be searching for a permutation.

In this combination we have three friends sitting two at a time.

C(3, 2) = 3 \times 2 = 6 possible combinations

That means that the friends rode the Ferris wheel SIX times in a row! Wow! No wonder they were dizzy!

Vocabulary

Here are the vocabulary words that are found in this lesson.

Probability
the chances or likelihood that an event will happen.
Outcome
the end result
Tree Diagram
a visual way of showing options and variables in an organized way. The lines of a tree diagram look like branches on a tree.
Combination
an arrangement of options where order does not make a difference.
Permutation
an arrangement of options where order does make a difference.

Technology Integration

Khan Academy, Permutations

Khan Academy, Combinations

James Sousa, Permutations

James Sousa, Combinations

Time to Practice

Directions: Design a tree diagram or use the Fundamental Counting Principle to determine each set of outcomes.

1. Jessica has three skirts and four sweaters. How many possible outfits can she arrange given her clothing?

2. Kim loves ice cream. She has the option of vanilla, chocolate or strawberry ice cream and she has different toppings to put on her ice cream cone. If she has sprinkles, hot fudge and nuts to choose from, how many different ice cream cones can she create with those toppings?

3. There are five possible surfboard designs and two possible colors. How many possible surfboards can be created from these options?

4. Team sweatshirts come in four colors and three sizes. How many sweatshirt outcomes are possible?

5. A diner offers six types of toast with either scrambled or fried eggs. How many breakfast options are there?

6. The same diner is offering a special that adds orange or apple juice with the eggs and toast. How many different breakfast options are there now?

7. If the diner adds in coffee as a beverage choice with the other options, how many different breakfast options can you have now?

8. If the diner also adds in a choice of bacon or sausage, how many different breakfast options do you have now?

9. An Italian restaurant offers penne pasta, shells or spaghetti with a choice of vegetable, meat or plain sauce. How many different pasta dishes are possible given these options?

10. If they also offer a choice of Italian bread or garlic bread, how many options are possible?

11. If they add in the choice of a Caesar salad or a tossed salad, how many meal options are there now?

12. If they offer a choice of ice cream or cheesecake with the meal, how many meal options are there now?

Directions: Figure out the following permutations.

13. P(5,2)

14. P(6, 3)

15. P(7, 2)

16. P(5, 4)

17. P(7, 3)

18. P(4, 4)

19. P(5, 3)

20. P(8, 4)

21. P(9, 4)

22. P(10, 3)

23. P(12, 2)

24. P(9, 3)

25. P(8, 6)

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