1. Fundamental Counting Principal
Good Morning happy Tuesday!

2. Counting Outcomes and Tree Diagrams
Tree diagrams are used to help visualize and count outcomes.
Example Tossing a coin 3 times

3. Tree Diagrams expanded
Tree diagrams can be used to organize information and count outcomes
Any path from left to right through the tree diagram is an outcome

4. Tree Diagrams Applied when actions are sequential
Root of the tree corresponds to the starting point
Branches connect the root to nodes representing different outcomes
Process continues until all stages are complete

5. Example
A fast food restaurant sells hot dogs, hamburgers, chicken sandwiches, and barbecue sandwiches. They offer as sides french fries, hushpuppies, or onion rings.
How many possible combinations are there?
Create a tree diagram to show this.

6. You try!
A mechanic offers three types oil changes (standard, synthetic, and high mileage), two types of wiper blades (low profile, and heavy use) and two types of mufflers (chrome and matte black).
What are the possible combinations?
Make a Tree Diagram to show possible outcomes.

7. Fundamental Counting Principle
Suppose that two events occur in order
The first can occur in m ways
The second can occur in n ways
Then the two events can occur in m x n ways

8. Example
An ice cream store offers three types of cones and 31 flavors. How many different single- cone ice-cream cones is it possible to buy at this store?
Types of cones? 3
Flavors of ice cream? 31

9. Solution 3 types of cones 31 flavors of ice cream
By the Fundamental Counting Principle, there are 3 x 31 = 93 ways of choosing a single-scoop ice- cream cone at this store

10. Example
In a certain state, automobile license plates display three letters followed by three digits. How many such plates are possible if repetition of the letter is allowed?
How many choices for each letter? 26
How many choices for each digit? 10

11. Solution 26 10
By the Fundamental Counting Principle, the number of possible license plates is…
26 x 26 x 26 x 10 x 10 x 10
17, 576, 000

12. How many choices for the first letter?
Example
In a certain state, automobile license plates display three letters followed by three digits. How many such plates are possible if repetition of the letters is not allowed?
How many choices for the first letter? 26
How many choices for the second letter? 25
How many choices for the third letter? 24

13. Solution 26 25 24 10
By the Fundamental Counting Principle, the number of possible license plates is…
26 x 25 x 24 x 10 x 10 x 10
15, 600, 000

14. Factorial Notation
Product of the first n natural numbers
Denoted by n! and is called n factorial
n! = n x (n – 1) x (n – 2) x … x 1
8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
0! = 1

15. Example
In how many different ways can a race with six runners be completed? Assume there is no tie.

16. Solution Six possible choices for first place
Five possible choices for second place
Four choices for third place and so on…
So, by the Fundamental Counting Principle the number of different ways the race can be completed is…
6 x 5 x 4 x 3 x 2 x 1 = 6! = 720