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Counting Principles and Tree DiagramsPowerPoint Presentation

Counting Principles and Tree Diagrams

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Fundamental counting principle

- The Fundamental Counting Principle states that if there are x ways to choose a first item and y ways to choose a second item, then there are x(y) ways to choose all items.

For example

- A telephone company is assigned a new area code and can issue new 7-digit phone numbers. All phone numbers are equally likely.
- Find the number of possible 7-digit phone numbers
- Use the Fundamental Counting Principle:
- 1st digit 2nd digit 3rd digit 4th digit 5th digit 6th digit 7th digit
- ? ?? ?? ? ?
- 10 10101010 10 10
- There are 10 choices for each digit (0-9), so there are
- 10(10)(10)(10)(10)(10)(10) = 10,000,000 phone number options

You try

- A telephone company is assigned a new area code and can issue new 7-digit phone numbers. All phone numbers are equally likely.
- Find the probability of a phone number that does not contain an 8.
- First, use the fundamental counting principle to find the number of phone numbers that do not contain an 8.
- 9(9)(9)(9)(9)(9)(9) = 4,782,969
- P(no 8) = 4,782,969 = .478
- 10,000,000

Using a tree diagram

- The Fundamental Counting Principle tells you only the number of outcomes in some experiments, not what the outcomes are. A tree diagram is a way to show all possible outcomes.
- For example:
- A pizza place specializes in two types of crust, sesame and plain, and sells five different toppings, onions, olives, ham, green peppers, and mushrooms. Create a tree diagram to show the options assuming there is only one type of crust with one topping.

Tree diagrams

- Using the Fundamental Counting Principle, we know that we should have 10 options: 2 pizza crusts, 5 toppings…2(5)=10

You try

- You are going on a trip. You can pack 2 pairs of pants, 3 shirts, and 2 sweaters for your vacation. Use a tree diagram to show all outfit options you can make if each outfit consists of a pair of pants, a shirt, and a sweater.
- There are 12 total outfits to choose from
- 2(3)(2) = 12

The addition counting principle

- If one group contains x objects and a second group contains y objects, and the groups have no objects in common, then there are x + y options.
- For exampleHow many items can you choose from Bergen’s Deli menu?
- None of the lists contains identical items, so use the Addition Counting Principle.
- Total Choices = Sandwiches + Salads + Soups
- T = 4 + 3 + 3
- There are 10 total items to choose from

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