1. A tree diagram is a segmented graph in the shape of a tree in which no branch leads from any vertex back to itself. Each path through it represents a mutually exclusive event. Tree Diagram

2. Calvary Christian Academy is having an election of student officers. Three students are running for president—Juan, Pam, and Jeff. There are two candidates for vice president— Doyle and Julianne. How many different ways are there to fill the offices?

3. Example 1 Find the number of ways that a student can select a two- digit number if the first digit must be odd and the second digit must be less than five. possible first digit—1, 3, 5, 7, 9 possible second digit— 0, 1, 2, 3, 4

4. 0 1 2 3 4 1133557799

5. 30, 31, 32, 33, 34 There are 25 different two-digit numbers. 50, 51, 52, 53, 54 70, 71, 72, 73, 74 90, 91, 92, 93, 94 10, 11, 12, 13, 14

6. Make a tree diagram to find the number of combinations of three pairs of pants, three coats, and four shirts. 36 Example

7. Make a tree diagram to find the number of possible milkshakes that could be ordered if chocolate and vanilla shakes are available in small, medium, and large. 66 Example

8. Make a tree diagram to find the number of ways to make fifty cents in change using nickels, dimes, and quarters. 10 Example

9. Fundamental Principle of Counting If there are p ways that a first choice can be made and q ways that a second choice can be made, then there are p × q ways to make the first choice followed by the second choice.

10. Reid has five dress shirts and four ties. How many different shirt-and-tie combinations are possible? 5 × 4 = 20 Example 2

11. How many different two- digit counting numbers can be formed if the first digit must be a nonzero even digit and the second digit must be less than seven but greater than zero? Example 3

12. There are four choices (2, 4, 6, 8) for the first digit and six choices (1, 2, 3, 4, 5, 6) for the second digit. By the Fundamental Principle of Counting there are 4 x 6 = 24 such numbers.

13. Mr. Dillard is buying a new car. He has the options given in the following table to choose from. How many different options does he have? If he chooses a white exterior, how many combinations does he have on the remaining options? Example 4

14. Interior red white black silver Transmission black blue gray AM/FM + CD + DVD automatic manual Package Exterior

15. How many different options does he have? If he chooses a white exterior, how many combinations does he have on the remaining options? How many different options does he have? If he chooses a white exterior, how many combinations does he have on the remaining options? 4 × 3 × 3 × 2 = 72 1 × 3 × 3 × 2 = 18

16. Use the Fundamental Principle of Counting to find the number of possible three-digit area codes if the first number cannot be 0 or 1. 800 Example

17. How many different license plates are possible if three letters must be followed by three numbers? 17,576,000 Example

18. How many different license plates are possible if none of the letters or numbers can repeat? 11,232,000 Example

19. How many ways can a family of four line up for a photograph? 24 Example

20. How many combinations are possible on a school locker if the lock consists of the numbers 1 to 40 and the combination is a three-digit sequence of numbers? 64,000 Example

21. How many combinations are possible if no two consecutive numbers are the same? 60,840 Example

22. How many ways can you seat five couples in a row of ten chairs, assuming, of course, that each couple is seated together? 3,840 Example