10-7

1. 10-7 Permutations Course 2 Warm Up Problem of the Day Lesson Presentation

2. Permutations 10-7 Course 2 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. Permutations 10-7 Course 2 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.) ?? ?? ?? ?? = 27 54 19 38 Possible answer: =

4. Permutations 10-7 Course 2 Learn to find the number of possible permutations and the probability that a specific permutation will occur.

5. Permutations 10-7 Course 2 Insert Lesson Title Here Vocabulary permutation factorial

6. Permutations 10-7 Course 2 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. Permutations 10-7 Course 2 Additional Example 1A: Using a List to Find Permutations A. In how many ways can you arrange the letters, A, B,and T ? Use a list to find the possible permutations. There are 6 ways to order the letters.

8. Permutations 10-7 number of word arrangements total number of arrangements P(word) = Course 2 Additional Example 1B: Using a List to Find Permutations B. If the order is chosen randomly, what is the probability that an arrangement will be a word? Of the 6 possible permutations, only 2 are words (B, A, T) and (T, A, B). 2 6 1 3 = = 1 3 The probability that an arrangement is a word is .

9. Permutations 10-7 Course 2 Insert Lesson Title Here Try This: Example 1A A. In how many ways can you arrange the colors red, orange, blue? Use a list to find the possible permutations. red, orange, blue red, blue, orange orange, red, blue orange, blue, red blue, orange, red blue, red, orange There are 6 ways to order the colors.

10. Permutations 10-7 number of word arrangements total number of arrangements P(word) = Course 2 Insert Lesson Title Here Try This: Example 1B B. If the order is chosen randomly, what is the probability that the first letters of the color arrangements will be a word? Of the 6 possible permutations, only three are words (orb), (bro), and (rob). 3 6 1 2 = = 1 2 The probability that an arrangement is a word is .

11. Permutations 10-7 Course 2 By making an organized list, you can find the possible permutations as well as the number of permutations. You can use the Fundamental Counting Principle to find only the number of permutations.

12. Permutations 10-7 Course 2 Additional Example 2: Using the Fundamental Counting Principle to Find the Number of Permutations Mary, Rob, Carla, and Eli are lining up for lunch. In how many different ways can they line up? What is the probability that any one of the permutations will be chosen at random? Once you fill a position, you have one less choice for the next position. 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 = 24 Multiply. There are 24 different ways the students can line up for lunch. The probability that any one of the permutations will be chosen is . 1 24

13. Permutations 10-7 Course 2 Insert Lesson Title Here Try This: Example 2 Find the number of ways you can rearrange the letters in the name Sam. What is the probability of any of the permutations being selected at random? 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 = 6 Multiply. There are 6 different ways the letters in the name Sam can be arranged. The probability that any one of the permutations will be chosen is . . 1 6

14. Permutations 10-7 Course 2 Multiplying 3 · 2 · 1 is called 3 factorial and is written as “3!” You can find the factorialof a whole number by multiplying all the whole numbers except zero that are less than or equal to the number. 3! = 3 · 2 · 1 = 6 6! = 6 · 5 · 4 · 3 · 2 · 1 = 720 You can use factorials to find the number of permutations in a given situation.

15. Permutations 10-7 Course 2 Additional Example 3: Using Factorials to Find the Number of Permutations In how many ways can Shellie line up 8 books on a shelf? 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.

16. Permutations 10-7 Course 2 Insert Lesson Title Here Try This: Example 3 In how many ways can Sally line up 5 pictures on a desk? Number of permutations = 5! = 5 · 4 · 3 · 2 · 1 = 120 There are 120 different ways for Sally to line up 5 pictures on a desk.

17. Permutations 10-7 Course 2 Insert Lesson Title Here Lesson Quiz 1. How many ways can Anna, Barbara, and Cara sit in a row? 2. If you choose one of the orders at random, what is the probability that Anna will be in the front? Find the number of permutations of each situation. 3. How many different ways could 4 people enter a roller-coaster car? 4. How many different ways could 6 basketball players sit on the bench while waiting to be announced at the beginning of a game? 6 1 3 24 720