Warm Up Problem of the Day Lesson Presentation Lesson Quizzes

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

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: =

Learn to find the number of possible permutations.

Vocabulary permutation factorial

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.

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

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. 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. There are 6 ways to order the colors.

You can use the Fundamental Counting Principle to find the number of permutations.

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 for lunch? 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 Fundamental Counting Principle states that you can find the total number of outcomes by multiplying the number of outcomes for each separate experiment. Remember!

Check It Out: Example 2 How many different ways can you rearrange the letters in the name Sam? Once you fill a position, you have one less choice for the next position. 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.

zero that are less than or equal to the number. 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.

Additional Example 3: Using Factorials to Find the Number of Permutations How many different orders are possible for Shellie to 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.

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.

Lesson Quiz for Student Response Systems Lesson Quizzes Standard Lesson Quiz Lesson Quiz for Student Response Systems 16 16 16 16

Lesson Quiz 1. In how many different ways can Anna, Barbara, and Cara sit in a row? 3. In how many different ways could 4 people enter a roller-coaster car? 4. 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

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

Lesson Quiz for Student Response Systems 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 19 19 19 19

Lesson Quiz for Student Response Systems 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 20 20 20 20