The Fundamental Counting Principle

1. The Fundamental Counting Principle MATH 102 Contemporary Math S. Rook

2. Overview • Section 13.2 in the textbook: • The fundamental counting principle • Slot diagrams

3. The Fundamental Counting Principle

4. The Fundamental Counting Principle • Recall that when situations can be broken up into stages, we used a tree diagram to illustrate all possibilities • Can become cumbersome if we have many stages or if one stage has many possibilities • Instead, observe the pattern that results when we use a tree diagram: • e.g. A restaurant offers 2 appetizers, 3 entrées, and 2 desserts. How many different meals can be created if a meal consists of one appetizer, one entrée, and one dessert?

5. The Fundamental Counting Principle (Continued) • This problem has how many stages/phases? • How many choices do we have for each stage/phase? • What is the relationship between the number of choices for each stage/phase and the number of possible meals? • Fundamental Counting Principle: If a situation can be broken up into stages where the first stage has a choices, the second stage has b choices, the third stage has c choices, etc, then the number of total possibilities is a x b x c x … • Do not forget to account for replacement if necessary

6. The Fundamental Counting Principle (Example) Ex 1: The board of an Internet start-up company has seven members. If one person is to be in charge of marketing and another in charge of research, in how many ways can these two positions be filled?

7. The Fundamental Counting Principle (Example) Ex 2: Elaine is building a home theater system consisting of a tuner, an optical disc player, speakers, and a high-definition TV. If she can select from four tuners, eight speakers, three optical disc players, and five TVs, in how many ways can she configure her system?

8. The Fundamental Counting Principle (Example) Ex 3: Determine the total number of possibilities when an 8-sided die and 12-sided die are rolled simultaneously.

9. Slot Diagrams

10. Slot Diagrams • A slot diagram is a visual way to illustrate the Fundamental Counting Principle • Useful in complex situations such as no repetition • Draw blanks for each stage of counting • Write number of possibilities in the blank • Multiply the possibilities together for all stages • May have to examine sub problems before applying the Fundamental Counting Principle

11. Slot Diagrams (Example) Ex 4: In a certain state, license plates currently consist of two letters followed by three digits. a) How many such license plates are possible if repeating is allowed? b) How many such license plates are possible if repeating is not allowed?

12. Slot Diagrams (Example) Ex 5:To enter a vault, an authorized user must provide a 5 digit numeric PIN. How many different PINs exist if: a) Digits are allowed to repeat? b) Digits are not allowed to repeat? c) The first digit is odd, the second digit is even, and the rest of the digits are not allowed to repeat? d) The PIN must be at least 59000 and digits are allowed to repeat?

13. Slot Diagrams (Example) Ex 6:Assume that we wish to seat three men: Alex, Carl, and Frank and three women: Bonnie, Daria, Edith in a row of six chairs. In how many ways can we seat everyone if: a) All the men sit together and all the women sit together? b) All the women sit together, but no woman sits on an end seat? c) Alex and Edith sit next to each other?

14. Slot Diagrams (Example) Ex 7: Assume we wish to seat two men: Alex and three women: Bonnie, Daria, and Edith in a row of six chairs. In how many ways can we seat everyone if the men sit together, the women sit together AND there is exactly one empty seat separating the two groups?

15. Summary • After studying these slides, you should know how to do the following: • Use the Fundamental Counting Principle to solve a variety of problems • Additional Practice: • See problems in Section 13.2 • Next Lesson: • Permutations & Combinations (Section 13.3)