1. Warm-Up Problem Can you predict which offers more choices for license plates? Choice A: a plate with three different letters of the alphabet in any order Choice B: a plate with four different nonzero digits in any order. By the end of class, you should be able to successfully answer this question.

2. Example 1 Eight pieces of paper are numbered from 1 to 8 and placed in a box. One piece of paper is drawn from the box, its number is written down, and the piece of paper is replaced in the box. A second piece of paper is drawn from the box, and its number is written down. If the two numbers are added together, how many different ways can a total of 12 be obtained?

3. Example 2 Eight pieces of paper are numbered from 1 to 8 and placed in a box. One piece of paper is drawn from the box and its number is written down. A second piece of paper is drawn from the box, and its number is written down. If the two numbers are added together, how many different ways can a total of 12 be obtained?

4. Example 3 You have three books that you want to place on a shelf. How many different ways can you arrange the books on the shelf?

5. The Counting Principle Let E 1 and E 2 be two events. The first event E 1 can occur m different ways. The second event E 2 can occur n different ways. The number of ways that the two events can occur is m times n.

6. Example 3 again. You have three books that you want to place on a shelf. How many different ways can you arrange the books on the shelf? 6 different ways to place 3 books on a shelf.

7. Example 4 You have five books and you want to place three of them on a shelf. How many different ways can you arrange three of the five books on the shelf? 60 different ways

8. This is the number of different arrangements or permutations of 5 things taken 3 at a time. This can be written as P(5,3). Remember: Permutations of n Elements taken r at a time is found by Key Words: arrangements, schedule, order

9. Example 5 Eight horses are running in a race. In how many different ways can these horses come in first, second, and third? Assume there are no ties. Using the Counting Principle: 8(7)(6)=336. Using Permutations:

10. Example 6 Your teacher gives you a list of three books. You must choose two of the books on the list to read. How many different pairings can you choose from?

11. Combinations When you choose r elements from a large set of n elements and ORDER DOES NOT MATTER, You are finding the number of combinations of n elements taken r at a time. Key Words: group, committee, selection, sample

12. Example 7 How many different committees of 3 people can be chosen from a group of 8 people? Since order does not matter, use combinations 56 committees

13. Warm-Up Problem Can you predict which offers more choices for license plates? Choice A: a plate with three different letters of the alphabet in any order Choice B: a plate with four different nonzero digits in any order. Alphabet – 15,600 Numbers - 3024