Algebra 1 Glencoe McGraw-Hill JoAnn Evans

Math 8H

12-2Counting Outcomes

Tree diagrams are a tool used to count the number of possible outcomes in a sample

space.

A tree diagram starts with one item, then branches into two or

more. Those branches each branch into two or more, and so on. The diagram resembles a tree, with a

trunk and multiple branches.

A one-topping pizza can be ordered with a choice of sausage, pepperoni, or mushrooms, a choice of thin or pan crust, and a choice of medium or

large size.

Toppings Crust Size Outcomes

Sausage

Pepperoni

Mushrooms

thin

pan

thin

pan

thin

pan

mediumlargemediumlargemediumlargemediumlargemediumlargemediumlarge

STMSTLSPMSPLPTMPTLPPMPPLMTMMTLMPMMPL

The Redwood coed volleyball team played 3 games against Los Cerritos Middle School. Show the different records the RMS team could have.

Game 1 Game 2 Game 3 Outcomes

win

lose

win

losewin

lose

winlosewinlosewinlosewinlose

WWWWWLWLWWLLLWWLWLLLWLLL

The number of possible outcomes can also be found without constructing a tree diagram.

Instead you can use the

which takes less time.

If an event M can occur in m ways and is followed by an event N that

can occur in n ways, then the event M followed by event N can

occur inm n ways.

Look at the tree diagram you made for the pizzas.

topping choices

3

crust choices

2

size choices

2

# of pizza outcomes

12

Look at the tree diagram you made for the games.

win/lose choices

2

win/losechoices

2

win/lose choices

2

# of outcomes

8

A sub sandwich restaurant offers four types of sub sandwiches, three different types of potato chips, five types of bread, and six different beverages.

How many different sandwich and drink combinations can you order?

# sub # chip # bread # beverage # outcomes

choices choices choices choices

4 3 5 3606

You could make a tree diagram to show the number of combinations, but it would take a

long time compared to the use of the Fundamental Counting Principle.

When Lindsay went on vacation she packed a variety of clothes. How many outfits were

possible for her to wear if she could choose one from each of four shirts, three pairs of pants, two

pairs of shoes and two jackets?

# shirt # pant # shoe # jacket # outfits choices choices choices choices

4 3 2 482

Lindsay could make 48 different outfits to wear from the clothes she

packed.

Ed and Fred went to an arcade that had 9 different games. In how many different orders

can they play the games if the play each one only once?

n 9 8 7 6 5 4 3 2 1

n 362,880

•Ed and Fred have nine games to choose from to play

first.•After choosing a game to play first, there are eight games left to choose from to play second.

•There would then be seven choices to play third.•This process will continue until all the games have been played.

There are 362,880 different orders.

9! 9 8 7 6 5 4 3 2 1

This is also known as a factorial, written as 9!

Factorials are very easy things.

Factorials are just products, indicated by an exclamation mark.

For example, “six factorial" is written as 6! and means

In general, n! means the product of all the whole numbers from n to 1.

6 5 4 3 2 1.

n! n n 1 n 2 ... 3 2 1

If Ed and Fred only have enough tokens to play 6 of the 9 different games, how many ways can they do

this?

There are still 9 choices for the first game, 8 choices for the second game, and so on, down

to four choices for the 6th game.

n 9 8 7 6 5 4

n 60,480 ways

Students at Thousand Oaks HS can choose class rings in one of each of 8 styles, 5 metals, 2 finishes, 14

stones, 7 cuts of stone, 4 tops, 3 printing styles, and 30 inscriptions. How many choices are there for a

class ring?8 5 2 14 7 4 3 30

2,822,400 choices

If a student narrows the choice to 2 styles, 3 metals, 4 cuts of stone, and 5 inscriptions (and has already

made the remaining decisions), how many choices are there for a class ring?

2 3 4 5 120 choices

In 1963 the US Postal Service instituted the use of five-digit ZIP codes to expedite mail delivery. In a ZIP code, the first digit corresponds to one of 10

national regions. The second and third digits form a number from 01 to 99 that corresponds to a metropolitan area. The last two digits form a number from 01 to 99 that corresponds to an

individual post office or zone. How many different 5-digit ZIP codes are possible?

10 99 99

98,010 ZI Pcodes

How many different outcomes are available for a four-digit number if the first digit must be even, the second digit must be odd, and the third and fourth

digits can be anything?

How many even digits are there?

Four (2, 4, 6, 8)

How many odd digits are there?

Five (1, 3, 5, 7, 9)

How many total digits are there?

Ten (0,1,2,3,4,5,6,7,8,9)

4 5 10 10 2000 possiblenumbers