1. Organized Counting Tree Diagrams Fundamental Counting Principle
Additive Counting Principle

2. Choices Choices Choices
Let’s take a Trip!
We are Travelling to __________ and we have a choice of two different flights from London, England. In order to get to London from Toronto you have a choice of 3 different flights (A,B,C). But, how are you getting to the airport? (Taxi, Bus, Drive).
So, how many ways can you get there?

3. Tree Diagrams
Each stage of the trip has multiple choices that branch off into different paths

4. Example
You are making a yummy sandwich and you have a choice of white or brown bread. On your sandwich you can have only one of the following; ham, chicken or beef. To dress the sandwich you can use mustard or mayonnaise. How many different kinds of sandwiches can you make?

5. Total of 12 Different Sandwiches White Ham Mustard Mayo Chicken Beef
Brown
Total of
12 Different Sandwiches

6. Multiplicative Counting Principle
These types of problems hold a fundamental counting principle that is multiplicative
If the problem has stages with multiple choices, then the total will be all the different choices in each stage multiplied together.
There are 2 types of Bread
There are 3 types of Meat
There are 2 types of Dressing
2x3x2 = 12

7. Additive Counting Principle
If a problem has multiple choices for different situations simply add each count together.
Also called the Rule of Sum

8. Example
Sailing ships use signal flags to send messages with 4 different flags. You must use a minimum of 2 flags for each message and can’t repeat flags How many different messages are there?

9. You can either fly 2 flags, 3 flags or 4 flags at a time
These actions can’t happen at the same time. ( you can’t fly 2 flags and be flying 4 flags at the same time)
These events are called MUTUALLY EXCLUSIVE events. They are separate actions that can’t happen at the same time.

10. Back to Example
Sailing ships use signal flags to send messages with 4 different flags. You must use a minimum of 2 flags for each message and can’t repeat flags. How many different messages are there?
2 Flag Message X3 =12
3 Flag Message X3X2 = 24
4 Flag Message X3X2X1=24
Add them all together
= 60 different possible messages

11. Example
How many ways can you arrange 8 people in line for a photo? BUT Matt and Kris just broke up and can’t be beside each other

12. But....
Sometimes it is easier to solve a problem indirectly. In this case we can find out how many ways it takes to put Matt and Kris beside eachother, and subtract it from the total possible matches
This is commonly called the BACK DOOR METHOD

13. Total Possible arrangements Put Matt and Kris beside each other
You can also look at pulling a left or a right and then there are is only 2 choices for each and only 2 of those are pairs so then you can multiply that by four for the other shoes that are there, giving you 8 pairs again to subtract from that total
Total arrangements – arrangements with them together
= arrangements of them apart =

14. Key Concepts Multiplicative Counting Principle
Multiply choices for each stage
Additive Counting Principle
Add up each mutually exclusive situation together
Back Door Method
Determine the opposite and subtract from total

15. See Handout Working with each other complete the handout given
Create a rule for the Fundamental Counting principle (multiplicative) and the Additive Counting Principle in the boxes provided
Complete the questions on the back and we will take them up in class

16. Homework
Page 229 # 3,5 – 17 (do not do #7), 24