1. 10-8 Permutations Warm Up Warm Up Lesson Presentation Lesson Presentation Problem of the Day Problem of the Day Lesson Quizzes Lesson Quizzes

2. 10-8 Permutations Warm Up 1. How many 2-side-dish meals can be made from 6 choices of side dishes? 2. Kim has shorts in blue, black, and tan. She has shirts in blue, yellow, red, and green. How many different combinations can she make? 3. If you go to the movies and are allowed to get 2 snacks and there are 9 snacks to choose from, how many combinations are there to pick from? 15 12 36

3. 10-8 Permutations Problem of the Day Replace each ? with a different digit from 1 through 9 to make a proportion. (Hint: The digits are not being multiplied.) Possible answer: ?? = 27 54 = 19 38

4. 10-8 Permutations I can find the number of possible permutations.

5. 10-8 Permutations Vocabulary permutation factorial

6. 10-8 Permutations An arrangement of objects or events in which the order is important is called a permutation. You can use a list to find the number of permutations of a group of objects.

7. 10-8 Permutations In how many ways can you arrange the letters A, B, and T ? Additional Example 1: Using a List to Find Permutations Use a list to find the possible permutations. There are 6 ways to order the letters. T, B, AB, T, AA, T, B T, A, BB, A, TA, B, T

8. 10-8 Permutations Check It Out: Example 1 In how many ways can you arrange the colors red, orange, blue? Use a list to find the possible permutations. There are 6 ways to order the colors. red, orange, blue red, blue, orange orange, red, blue orange, blue, red blue, orange, red blue, red, orange List all permutations beginning with red, then orange, and then blue.

9. 10-8 Permutations You can use the Fundamental Counting Principle to find the number of permutations.

10. 10-8 Permutations Mary, Rob, Carla, and Eli are lining up for lunch. In how many different ways can they line up for lunch? Additional Example 2: Using the Fundamental Counting Principle to Find the Number of Permutations There are 4 choices for the first position. There are 3 remaining choices for the second position. There are 2 remaining choices for the third position. There is one choice left for the fourth position. 4 · 3 · 2 · 1 There are 24 different ways the students can line up for lunch. Multiply.= 24 Once you fill a position, you have one less choice for the next position.

11. 10-8 Permutations The Fundamental Counting Principle states that you can find the total number of outcomes by multiplying the number of outcomes for each separate experiment. Remember!

12. 10-8 Permutations Check It Out: Example 2 How many different ways can you rearrange the letters in the name Sam? There are 3 choices for the first position. There are 2 remaining choices for the second position. There is one choice left for the third position. 3 · 2 · 1 There are 6 different ways the letters in the name Sam can be arranged. Multiply.= 6 Once you fill a position, you have one less choice for the next position.

13. 10-8 Permutations A factorial of a whole number is the product of all the whole numbers except zero that are less than or equal to the number. “3 factorial” is 3! = 3 · 2 · 1 = 6 “6 factorial” is 6! = 6 · 5 · 4 · 3 · 2 · 1 = 720 You can use factorials to find the number of permutations.

14. 10-8 Permutations How many different orders are possible for Shellie to line up 8 books on a shelf? Additional Example 3: Using Factorials to Find the Number of Permutations Number of permutations = 8! = 8 · 7 · 6 · 5 · 4 · 3 · 2 · 1 = 40,320 There are 40,320 different ways for Shellie to line up 8 books on the shelf.

15. 10-8 Permutations Check It Out: Example 3 How many different orders are possible for Sherman to line up 5 pictures on a desk? Number of permutations = 5! = 5 · 4 · 3 · 2 · 1 = 120 There are 120 different ways for Sherman to line up 5 pictures on a desk.

16. 10-8 Permutations Standard Lesson Quiz Lesson Quizzes Lesson Quiz for Student Response Systems

17. 10-8 Permutations Lesson Quiz 1. In how many different ways can Anna, Barbara, and Cara sit in a row? 2. In how many different ways could 4 people enter a roller-coaster car? 3. How many different orders are possible for 6 basketball players to sit on the bench while waiting to be announced at the beginning of a game? 6 24 720

18. 10-8 Permutations 1. Identify the number of ways you can arrange the letters in the word “MATH”. A. 4 B. 6 C. 16 D. 24 Lesson Quiz for Student Response Systems

19. 10-8 Permutations 2. In how many different ways can you arrange the numbers 1, 3, 5, 7, 9 to make a 5-digit number without any repetitions? A. 5 B. 25 C. 120 D. 720 Lesson Quiz for Student Response Systems

20. 10-8 Permutations 3. Janet has 9 antique pieces. In how many different ways can she arrange them on a shelf? A. 362,880 B. 40,320 C. 81 D. 9 Lesson Quiz for Student Response Systems