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Tree Diagrams Fundamental Counting Principle Additive Counting Principle Created by: K Wannan Edited by: K Stewart

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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?

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Each stage in the sandwich making process has multiple choices. Together, different paths are made that show all of the possible outcomes (sandwiches)

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WhiteHamMustardMayoChickenMustardMayoBeefMustardMayoBrownHamMustard Mayo Chicken MustardMayoBeefMustardMayo Total of 12 Different Sandwiches Is there a faster way to find this result than counting results in a tree diagram?

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Why can we multiply the number of options in each stage to get the total number of possible outcomes (in this case, the total number of possible sandwiches)? There are 2 types of Bread There are 3 types of Meat There are 2 types of Dressing 2x3x2 = 12

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At lunch in a restaurant you are given a choice of either a soup or a salad as an appetizer. There are five soups and four salads to choose from. How many different options are there for appetizers?

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By counting, we can see that there are nine choices. Is there a faster way than drawing an image and counting?

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Why can we add the number of options in each stage to get the total number of possible outcomes (in this case, the total number of possible appetizers)? There are five soups and four salads 5 + 4 = 9

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Sailing ships use signal flags to send messages. There are four different flags. Messages are sent using at least two of these flags for each message. Flags cannot be repeated. How many different messages are possible?

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You can either fly two flags or three flags or four flags at a time. These actions cant happen at the same time. ( it isnt possible to fly only two flags and also be flying four flags simultaneously!) These events are called MUTUALLY EXCLUSIVE events. They are separate actions that cant happen at the same time.

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Sailing ships use signal flags to send messages. There are four different flags. Messages are sent using at least two of these flags for each message. Flags cannot be repeated. How many different messages are possible? Determining the number of possible messages for 2, 3 and 4 flags uses the multiplicative counting principle. Why? 2 Flag Message 4X3 =12 3 Flag Message 4X3X2 = 24 4 Flag Message 4X3X2X1=24 The total number of possible messages in the three possible situations uses the additive counting principle. Why? 12 + 24 + 24 = 60 different possible messages

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Steph has four pairs of shoes in her gym bag. How many ways can she pull out an UNMATCHED (left and right) pair of shoes? Sometimes it is easier to solve a problem indirectly. In this case we can find out how many ways it takes to find a MATCHED pair, and subtract it from the total possible matches This type of indirect reasoning is commonly called the BACK DOOR METHOD

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Total Possible Pairs Pull #1 Pull #2 8 shoes x 7 shoes = 56 A Matched Pair Pull #1 can be any one of the shoes (left or right of one of the four pairs) and then Pull #2 has only one possibility for a match Totalling 8 ways to draw a matched pair of shoes. 56 - 8 = 48 Ways that Steph can pull an UNMATCHED pair of shoes Pull #1 could be Pull #2 has to be (to match) L1R1 L2R2 L3R3 L4R4 R1L1 R2L2 R3L3 R4L4

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Multiplicative Counting Principle multiply choices for each stage when its a multi-step situation where every choice in one stage is possible for each of the choices in the previous stage, e.g. There are four choices for appetizer and three choices for dessert. Choose an appetizer and a dessert. Additive Counting Principle add number of choices in each situation together when situations are mutually exclusive e.g. There are four choices for appetizer and three choices for dessert. Choose either an appetizer or a dessert (not both). Indirect Reasoning Back Door Method Determine the number of favourable outcomes for the opposite event and subtract from total possible number of outcomes.

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