1. 0.4 Counting Techniques

2. Fundamental Counting Principle TWO EVENTS:If one event can occur in m ways and another event can occur in n ways, then the number of ways that both events can occur is m  n. (If one event can occur in 3 ways and another event can occur in 6 ways, then both events can occur in 3  6 = 18 ways.) THREE OR MORE EVENTS: The fundamental counting principle “extended”. If 3 events can occur in m, n, and p ways, then the number of ways that all three events can occur is m  n  p. (If three events can occur in 3, 4, and 8 ways, then all three events can occur in 3  4  8 = 96 ways.)

3. Example: 1) At Oswego East High School, there are 603 freshmen, 470 sophomores, 446 juniors, and 292 seniors. In how many different ways can a committee of 1 person from each class be chosen? 2) Mr. and Mrs. Cal Q. Leight go out to dinner after a long day. At the restaurant they go to, they each have a choice of 8 different entrees, 2 different salads, 12 different soft drinks, and 6 different desserts. In how many ways can Mr. C choose 1 salad, 1 entrée, 1 soft drink, and 1 dessert?

4. Fundamental Counting Principle with Repetition Examples: 1) The standard configuration of an Illinois license plate use to be 3 letters followed by 3 digits. a) How many different license plates are possible if letters and digits can be repeated? b) How many different license plates are possible if letters and digits can not be repeated? 2) A multiple choice test has 10 questions with 4 multiple choices for each question. In how many different ways could you complete the test?

5. Factorial Counting Problems also involve determining the number of different objects in a certain order. This is called a permutation.

6. Permutations Permutation: an ordering of n objects, where the order of the objects does matter. EX: There are 6 permutations of the letters A, B, and C: ABC, ACB, BCA, BAC, CAB, and CBA. Since there are 3 choices for the 1 st letter, 2 choices for the 2 nd, and 1 choice for the 3 rd, there are 3∙2∙1 = 6 ways to arrange the letters. (Remember 3∙2∙1 can be written 3!) In general, the number of permutations of n objects is n!

7. 1) 10 skiers are competing in the final round of the Olympic freestyle skiing aerial competition. a) In how many different ways can the skiers finish the competition? (Assume there are no ties) b) In how many different ways can 3 of the runners finish first, second, and third to win the gold, silver, and bronze? 2) You have homework assignments from 5 different classes to complete this weekend. a) In how many different ways can you complete the assignments? b) In how many different ways can you choose 2 of the assignments to complete first and last?

8. 3) There are 8 movies you would like to see that are currently showing in theaters! a) In how many different ways can you see all of the 8 movies? b) In how many ways can you choose a movie to see this Saturday and one to see this Sunday?

9. Permutations of n objects taken r at a time The number of permutations of r objects taken from a group of n distinct objects is denoted by Examples: 1) There are 12 books on the summer reading list. You want to read some or all of them. In how many orders can you read… a) 4 of the books? b) all 12 of the books?

10. 2) You are considering 11 different colleges. Before you decide to apply to the colleges you want to visit some or all of them. In how many orders can you visit… a)5 of the colleges? b)11 of the colleges? 3) There are 9 players on a baseball team. a) In how many ways can you choose the batting order for all 9 of the players? b) In how many ways can you choose a pitcher, catcher, and shortstop from the 9 players?

11. Permutations with Repetition Permutations that are not distinguishable, meaning you can’t tell one permutation or arrangement from another one in the list, are often found with repeating objects…like the letters in the word WOW. WOW OWW WWO Only 3 are distinguishable without color (WOW, OWW, WWO)

12. In general, the number of distinguishable permutations of n objects where one object is repeated q 1 times, another is repeated q 2 times, and so on is: Examples: Find the # of distinguishable permutations of the letters. 1) SUMMER 2) WATERFALL 3) ILLINOIS