Subway and Starbucks advertise over 10,000 combinations of ways to customize sandwiches and drinks due to the many condiment and serving options. Companies can calculate combinations using (1) tree diagrams to show all outcomes or (2) the fundamental counting principle of multiplying the number of options at each choice point. Tree diagrams specifically show individual outcomes while the counting principle gives the total number of outcomes. Examples demonstrate using each method to find combinations in probability problems.

1. TheThe
CountingCounting
PrinciplePrinciple

2. Counting OutcomesCounting Outcomes
Have you ever seen or heard theHave you ever seen or heard the
Subway or Starbucks advertisingSubway or Starbucks advertising
campaigns where they talk about thecampaigns where they talk about the
10,000 different combinations of ways10,000 different combinations of ways
to order a sub or drink?to order a sub or drink?

3. Counting OutcomesCounting Outcomes
Have you ever seen or heard theHave you ever seen or heard the
Subway or Starbucks advertisingSubway or Starbucks advertising
campaigns where they talk about thecampaigns where they talk about the
10,000 different combinations of ways10,000 different combinations of ways
to order a sub or drink?to order a sub or drink?
When companies like these makeWhen companies like these make
these claims they are using all thethese claims they are using all the
different condiments and ways todifferent condiments and ways to
serve a drink.serve a drink.

4. Counting OutcomesCounting Outcomes
- These companies can use (2) ideas- These companies can use (2) ideas
related to combinations to make theserelated to combinations to make these
claims:claims:
(1) TREE DIAGRAMS(1) TREE DIAGRAMS
(2) THE FUNDAMENTAL(2) THE FUNDAMENTAL
COUNTING PRINCIPLECOUNTING PRINCIPLE

5. Counting OutcomesCounting Outcomes
(1) TREE DIAGRAMS(1) TREE DIAGRAMS
A tree diagram is a diagram used to showA tree diagram is a diagram used to show
the total number of possible outcomes inthe total number of possible outcomes in
a probability experiment.a probability experiment.

6. Counting OutcomesCounting Outcomes
(2) THE COUNTING PRINCIPLE(2) THE COUNTING PRINCIPLE
The Counting Principle uses multiplication ofThe Counting Principle uses multiplication of
the number of ways each event in anthe number of ways each event in an
experiment can occur to find the numberexperiment can occur to find the number
of possible outcomes in a sample space.of possible outcomes in a sample space.
http://http://
www.youtube.com/watch?v=8WdSJhEIrQk&swww.youtube.com/watch?v=8WdSJhEIrQk&s

7. Counting OutcomesCounting Outcomes
Example 1Example 1:: Tree Diagrams.Tree Diagrams.
A new polo shirt is released in 4 differentA new polo shirt is released in 4 different
colors and 5 different sizes. How manycolors and 5 different sizes. How many
different color and size combinationsdifferent color and size combinations
are available to the public?are available to the public?
Colors –Colors – (Red, Blue, Green, Yellow)(Red, Blue, Green, Yellow)
Styles –Styles – (S, M, L, XL, XXL)(S, M, L, XL, XXL)

8. The Tree DiagramThe Tree Diagram
RED YELLOWGREENBLUE
S
M
L
XL
XXL
S
M
L
XL
XXL
S
M
L
XL
XXL
S
M
L
XL
XXL

9. A Different WayA Different Way
Example 1Example 1:: The Counting Principle.The Counting Principle.
A new polo shirt is released in 4 differentA new polo shirt is released in 4 different
colors and 5 different sizes. How manycolors and 5 different sizes. How many
different color and size combinationsdifferent color and size combinations
are available to the public?are available to the public?
Colors –Colors – (Red, Blue, Green, Yellow)(Red, Blue, Green, Yellow)
Styles –Styles – (S, M, L, XL, XXL)(S, M, L, XL, XXL)

10. Counting OutcomesCounting Outcomes
Example 1Example 1:: The Fundamental CountingThe Fundamental Counting
Principle.Principle.
Answer.Answer.
Number ofNumber of Number ofNumber of Number ofNumber of
Possible ColorPossible Color Possible SizesPossible Sizes Possible Comb.Possible Comb.
44 xx 55 == 2020

11. Counting OutcomesCounting Outcomes
 Tree Diagrams and The FundamentalTree Diagrams and The Fundamental
Counting Principle are two differentCounting Principle are two different
algorithms for finding sample space ofalgorithms for finding sample space of
a probability problem.a probability problem.
 However, tree diagrams work betterHowever, tree diagrams work better
for some problems and thefor some problems and the
fundamental counting principle worksfundamental counting principle works
better for other problems.better for other problems.

12. So when should I use a tree diagram orSo when should I use a tree diagram or
the fundamental counting principle?the fundamental counting principle?
- A- A tree diagramtree diagram is used to:is used to:
(1) show sample space;(1) show sample space;
(2) count the number of preferred outcomes.(2) count the number of preferred outcomes.
- The- The fundamental counting principlefundamental counting principle cancan
be used to:be used to:
(1) count the total number of outcomes.(1) count the total number of outcomes.

13. Counting OutcomesCounting Outcomes
Example 2Example 2:: Tree Diagram.Tree Diagram.
Tamara spins a spinner twoTamara spins a spinner two
times. What is her chancetimes. What is her chance
of spinning a green on theof spinning a green on the
first spin and a blue on the second spin?first spin and a blue on the second spin?
You use a tree diagram because you want aYou use a tree diagram because you want a
specific outcome … not the TOTALspecific outcome … not the TOTAL
number of outcomes.number of outcomes.

14. Counting OutcomesCounting Outcomes
Example 2Example 2:: Tree Diagram.Tree Diagram.
Tamara spins a spinner twoTamara spins a spinner two
times. What is her chancetimes. What is her chance
of spinning a green on theof spinning a green on the
first spin and a blue on the second spin?first spin and a blue on the second spin?

15. Counting OutcomesCounting Outcomes
Example 3Example 3:: The Counting Principle.The Counting Principle.
If a lottery game is made up of threeIf a lottery game is made up of three
digits from 0 to 9, what is the totaldigits from 0 to 9, what is the total
number of outcomes?number of outcomes?
You use the Counting Principle because youYou use the Counting Principle because you
want the total number of outcomes. Howwant the total number of outcomes. How
many possible digits are from 0 to 9?many possible digits are from 0 to 9?

16. Counting OutcomesCounting Outcomes
Example 3Example 3:: The Fundamental CountingThe Fundamental Counting
Principle.Principle.
If a lottery game is made up of three digitsIf a lottery game is made up of three digits
from 0 to 9, what is the total number offrom 0 to 9, what is the total number of
possible outcomes?possible outcomes?
# of Possible# of Possible # of Possible# of Possible # of Possible# of Possible # of Possible# of Possible
DigitsDigits DigitsDigits DigitsDigits OutcomesOutcomes
10 x 10 x 10 =10 x 10 x 10 = 10001000
What chance would you have to win if you played one time?What chance would you have to win if you played one time?

17. Guided PracticeGuided Practice:: Tree or Counting Principle?Tree or Counting Principle?
(1) How many outfits are possible from a pair(1) How many outfits are possible from a pair
of jean or khaki shorts and a choice ofof jean or khaki shorts and a choice of
yellow, white, or blue shirt?yellow, white, or blue shirt?
(2) Scott has 5 shirts, 3 pairs of pants, and 4(2) Scott has 5 shirts, 3 pairs of pants, and 4
pairs of socks. How many different outfitspairs of socks. How many different outfits
can Scott choose with a shirt, pair ofcan Scott choose with a shirt, pair of
pants, and pair of socks?pants, and pair of socks?

18. Example 1Example 1
 You are purchasing a new car. Using theYou are purchasing a new car. Using the
following manufacturers, car sizes and colors,following manufacturers, car sizes and colors,
how many different ways can you select onehow many different ways can you select one
manufacturer, one car size and one color?manufacturer, one car size and one color?
Manufacturer: Ford, GM, ChryslerManufacturer: Ford, GM, Chrysler
Car size: small, mediumCar size: small, medium
Color: white(W), red(R), black(B), green(G)Color: white(W), red(R), black(B), green(G)

19. SolutionSolution
 There are three choices of manufacturer, twoThere are three choices of manufacturer, two
choices of car sizes, and four colors. So, thechoices of car sizes, and four colors. So, the
number of ways to select one manufacturer, onenumber of ways to select one manufacturer, one
car size and one color is:car size and one color is:
3 ●2●4 = 24 ways.3 ●2●4 = 24 ways.

20. Ex. 2 Using the Fundamental CountingEx. 2 Using the Fundamental Counting
PrinciplePrinciple
 The access code for a car’s security systemThe access code for a car’s security system
consists of four digits. Each digit can be 0consists of four digits. Each digit can be 0
through 9. How many access codes are possiblethrough 9. How many access codes are possible
if each digit can be repeated?if each digit can be repeated?