Tree Diagrams
Fundamental Counting Principle
Additive Counting Principle
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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?
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Each stage of the trip has multiple choices
that branch off into different paths
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?
Mustard
Ham
White
Chicken
Mayo
Mustard
Mayo
Beef
Mustard
Mayo
Ham
Mustard
Mayo
Brown
Chicken
Mustard
Mayo
Mustard
Beef
Mayo
Total of
12 Different
Sandwiches
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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
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If a problem has multiple choices for different
situations simply add each count together.
Also called the Rule of Sum
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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?
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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.
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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
3 Flag Message
4 Flag Message
4X3 =12
4X3X2 = 24
4X3X2X1=24
Add them all together
12 + 24 + 24 = 60 different
possible messages
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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
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
Total Possible arrangements
Put Matt and Kris beside each other
Total arrangements – arrangements with them together
= arrangements of them apart
40320 -10080 =
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Multiplicative Counting Principle
◦ Multiply choices for each stage
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Additive Counting Principle
◦ Add up each mutually exclusive situation together
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Back Door Method
◦ Determine the opposite and subtract from total
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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
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Page 229 # 3,5 – 17 (do not do #7), 24