0

2

4

6

8

10Won Lost

1 2 3 4

Year

Num

ber

of G

ames

Warm-Up

1) In which year(s) did the team lose more games than they won?

2) In which year did the team play the most games?

3) In which year did the team play ten games?

Tree Diagrams and Lists

Tree Diagrams: resembles the branches of a tree and show all possible outcomes by following each of the branches of the tree.

Lists: work best if a systematic way/plan is developed to list all possible outcomes.

Tree DiagramsTree diagrams allow us to

see all possible outcomes of an event (and calculate their probabilities …eventually)

This tree diagram shows the probabilities of results of flipping a coin three times.

The MHS cafeteria offers chicken or tuna sandwiches; chips or fruit; and milk, apple juice, or

orange juice. If you purchase one sandwich, one side item and one drink, how many different

lunches can you choose?

Sandwich(2) Side Item(2) Drink(3) Outcomes

chicken

tuna

There are 12 possible lunches.

chips

fruit

chips

fruit

apple juice orange juice milkapple juice orange juice milk

apple juice orange juice milkapple juice orange juice milk

chicken, chips, apple chicken, chips, orange chicken, chips, milkchicken, fruit, apple chicken, fruit, orange chicken, fruit, milk

tuna, chips, apple tuna, chips, orange tuna, chips, milktuna, fruit, apple tuna, fruit, orange tuna, fruit, milk

The Multiplication Counting Principle … an easier way!

If one event can occur in m ways and another event can occur in n ways, then the number of ways that both

events can occur together is m∙n. This principle can be extended to three or

more events.Basically…multiply the number of choices

you have at each stage.

So from the lunch example…

# sandwich choices x # side choices x # drink choices

Multiplication Counting Principle

At a sporting goods store, skateboards are available in 8 different deck designs. Each deck design is available with 4 different wheel assemblies. How many skateboard choices does the store offer?

32

The Multiplication Counting Principle Example:

How many different looks can you give Mr. Potato Head if we can choose from: 4 hats, 2 noses, 3 eyes, 2 shoes, and 5 ties?

240

How many ways can you make lunches out of a soup, a sandwich, some dessert, and a drink, given that there are 3 different soups to choose from, 4 kinds of sandwich, 4 desserts, and 5 drinks?

Multiplication Counting Principle

Multiplication Counting Principle

Rajohn’s aunt takes him to Wendy’s for lunch. She tells Rajohn he can get an

entrée, a side, and a drink. For the entrée, his choices are the 5 piece

nuggets, a spicy chicken sandwich, or a single. For sides: he can get fries, a side salad, potato, or chili. If Rajohn has 48

total lunch choices, how many DRINKS could Rajohn choose from?

4

The Addition Counting PrincipleA box contains 5 bags of milk

chocolate M&M’s, 5 bags of peanut M&M’s, 5 bags of sour skittles, 5 bags of regular skittles, 5 bags of chocolate covered raisins, and 5

bags of tropical skittles. How many bags are a variety of

skittles?

The Addition Counting Principle

If the outcome of interest can be divided into groups with no

possibilities in common, then the number of possibilities is the sum of the numbers of possibilities in

each group.

15

Many mp3 players can shuffle the songs that are played. Your mp3 currently only contains 8

songs (if you’re a loser). Find the number of orders in which the songs can be played.

There are 40,320 possible song orders.

In this situation it makes more sense to use the Fundamental Counting Principle.

• 7 • 6• 5 • 4 • 3• 2 •18 = 40,320

Factorial

EXAMPLE with Songs ‘eight factorial’

The product of counting numbers beginning at n and counting backward to 1 is written n! and it’s called n factorial.

factorial.

8! = 8 • 7 • 6 • 5 • 4 • 3 • 2 • 1 = 40,320

Calculators = the easy life!

FactorialSimplify each expression.

a. 4!

b. 6!

c. For the 8th grade field events there are five teams: Red, Orange, Blue, Green, and Yellow. Each team chooses a runner for lanes one through 5. Find the number of ways to arrange the runners.

4 • 3 • 2 • 1 = 24

6 • 5 • 4 • 3 • 2 • 1 = 720

= 5! = 5 • 4 • 3 • 2 • 1 = 120

The student council of 15 members must choose a president, a vice

president, a secretary, and a treasurer.

President Vice Secretary Treasurer Outcomes

There are 32,760 ways for choosing the class officers.

In this situation it makes sense to use the Fundamental Counting Principle.

15•14

13• •12=32,760

Cubes

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