10-8 Permutations Vocabulary permutation factorial

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.

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

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.

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

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.

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!

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.

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.

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.

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.