Transcript Toppings Crust Size Outcomes

Geometry
Counting 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
Sausage
Pepperoni
Mushrooms
Crust
thin
pan
thin
pan
thin
pan
Size
medium
large
medium
large
medium
large
medium
large
medium
large
medium
large
Outcomes
STM
STL
SPM
SPL
PTM
PTL
PPM
PPL
MTM
MTL
MPM
MPL
The girls volleyball team played 3 games against the
boys team. Show the different records the girls
team could have.
Game 1
win
lose
Game 2
win
lose
win
lose
Game 3
win
lose
win
lose
win
lose
win
lose
Outcomes
WWW
WWL
WLW
WLL
LWW
LWL
LLW
LLL
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 in
m•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/lose
choices

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
choices
4
# chip
choices

3
# beverage  # outcomes
choices
# bread
choices

5

6

360
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
choices
4
# pant
choices

3
# shoe
choices

2
# jacket
# outfits
choices 

2

Lindsay could make 48 different outfits
to wear from the clothes she packed.
48
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?
•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.
n  98 7  65 4 321
n  362,880
There are 362,880 different orders.
This is also known as a factorial, written as 9!
9!  9  8  7  6  5  4  3  2  1
Factorials are just products, indicated by
an exclamation mark.
For example, “six factorial" is written as
6! and means 6•5•4·3·2·1
In general, n! means the product of all the
whole numbers from n to 1.
n !  n   n  1    n  2   ...  3  2  1
STRANGE FACT: 0!=1 and so does 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 ZIP codes
How many different outcomes are available for a fourdigit 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 possible numbers