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Notes #21 The Fundamental Counting Principle

Notes #21 The Fundamental Counting Principle. Fundamental Counting Principle. GOAL: Use the Fundamental Counting Principle to determine the number of possible outcomes in a given situation. How to Apply the Fundamental Counting Principle ANS:

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Notes #21 The Fundamental Counting Principle

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  1. Notes #21The Fundamental Counting Principle

  2. Fundamental Counting Principle GOAL: Use the Fundamental Counting Principle to determine the number of possible outcomes in a given situation. How to Apply the Fundamental Counting Principle ANS: If you have 2 events: 1 event can occur “M” ways and another event can occur “N” ways, then the number of ways that both can occur is M∙N.

  3. Review of Using a Tree Diagram to represent of all possible choices. Ex 1: A man has 2 pair of jeans (Event 1) and 3 t-shirts (Event 2). How many different outfits can he put together? Using the Counting Principle to determine all possible choices. Event 1: (Jeans) Event 2: (t-shirts) Items: 2 pairs Items: 3shirts ANS: 2 pair of jeans x 3 t-shirts =6different outfits can be put together.

  4. Example 2Applying the Fundamental Counting Principle The Greasy Spoon Restaurant offers 6 appetizers and 14 main courses. In how many ways can a person order a two course meal?

  5. Set up: (4 Steps) • Specify the Events • Event 1: Appetizers b) Event 2: Main courses 2) Indicate the # of items per Event a. Event 1: 6 types of Appetizers b. Event 2: 14 main courses 3) Apply the CountingPrinciple 6 appetizers x 14 main courses = 84 4) Answer in a complete sentence ANS: Choosing from one of 6 appetizers and one of 14 main courses, a person has 84 ways to order a two-course meal.

  6. The Fundamental Counting Principle with More than Two Groups of Items • Definition 3 events can occur m, n, & p ways, then the number of ways all three can occur is m∙n∙p Section 11.1

  7. Example 3 The number of possible outfits from 2 pair of jeans, 3 T-shirts and 2 pairs of sneakers are:

  8. Set up (4 Steps): • List the Events a) Event 1: Jeans b) Event 2: T-Shirts c)Event 3: Sneakers 2) List # of Items per Event a) Jeans: 2 pair b) T-shirts: 3 shirts c) Sneakers: 2 pair 3) Applying the Counting Principle 2 pair of jeans x 3 t-shirts x 2 pair of sneakers = 12diff. outfits 4) Concluding Sentence: There are 12 different outfit that can be made. Section 11.1

  9. Example 4Options in Planning a Course Schedule Next semester, you are planning to take three courses – Math, English and Humanities. There are 8 sections of Math, 5 of English, and 4 of Humanities that you find suitable. Q: Assuming no scheduling conflicts, how many different three-course schedules are possible?

  10. Example 4Options in Planning a Course Schedule Q: Assuming no scheduling conflicts, how many different three-course schedules are possible? Solution: This situation involves making choices with three groups of items. MathEnglishHumanities {8 choices} {5 choices} {4 choices} ANS: There are 8 ∙ 5 ∙ 4 = 160 diff. three-course schedules.

  11. Example 5Telephone Numbers in the United States • Telephone numbers in the United States begin with three-digit area codes followed by seven-digit local telephone numbers. - Area codes and local telephone numbers cannot begin with 0 or 1. Q: How many different telephone numbers are possible? Section 11.1

  12. Telephone Numbers in the United States Q: How many different telephone numbers are possible? Solution: This situation involves making choices with ten groups of items. Here are the choices for each of the ten groups of items: CONDITION: Area codes and local telephone numbers cannot begin with 0 or 1. Area CodeLocal Telephone Number The total number of different telephone numbers is: 8 ∙ 10 ∙ 10 ∙ 8 ∙ 10 ∙ 10 ∙ 10 ∙ 10 ∙ 10 ∙ 10 = 6,400,000,000 8 8 10 10 10 10 10 10 10 10 Section 11.1

  13. Section 11.1

  14. Example 5Telephone Numbers in the United States Q: How many different telephone numbers are possible? Solution: This situation involves making choices with ten groups of items. Here are the choices for each of the ten groups of items: Area CodeLocal Telephone Number 8 10108101010101010 The total number of different telephone numbers is: 8 ∙ 10 ∙ 10 ∙ 8 ∙ 10 ∙ 10 ∙ 10 ∙ 10 ∙ 10 ∙ 10 = 6,400,000,000 Section 11.1

  15. Example 5 A Multiple-Choice Test • You are taking a multiple-choice test that has ten questions. Each of the questions has four answer choices, with one correct answer per question. If you select one of these four choices for each question and leave nothing blank, in how many ways can you answer the questions? • Solution: This situation involves making choices with ten questions: Question 1 Question 2 Question 3 ∙ ∙ ∙ Question 9 Question 10 {4 choices} {4 choices} {4 choices} {4 choices} {4 choices} The number of different ways you can answer the questions is: 4 ∙ 4 ∙ 4 ∙ 4 ∙ 4 ∙ 4 ∙ 4 ∙ 4 ∙ 4 ∙ 4 = 410 = 1,048,576 Section 11.1

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