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Permutations

Permutations What is a Permutation? - Have you ever been in an ice cream shop and wondered about all the different ways you could order three different scoops of ice cream? - A PERMUTATION is an arrangement or listing in which order IS important. -the act or process of changing the lineal order of an ordered set of objects -an ordered arrangement of a set of objects

Permutations Real World Example: Five students are finalists in the school spelling bee. How many ways can they finish first, second, and third?

Permutations Real World Example: Five students are finalists in the school spelling bee. How many ways can they finish first, second, and third? P(5,3) = 5 x 4 x 3 = 60 different ways Also written 5 P 3

Permutations How Do I Find The Value of A Permutation? - We calculate the value of a permutation in the following way: P(5,3) = 5 x 4 x 3 = 60 different ways Start with this number Count down this many numbers (1)(2)(3)

Permutations Example 1: Permutations. Find the value for P(5,2).

Permutations Example 1: Permutations. Find the value for P(5,2). P(5,2) = 5 x 4 = 20 Start with this number We are using this many numbers so we count down this many numbers (1)(2)

Permutations Example 2: Standing in Line. In how many different ways can Carlos, Sergio, Caleb, DeMoris, Eric, and Brayton stand in line?

Permutations Example 2: Standing in Line. In how many different ways can Carlos, Sergio, Caleb, DeMoris, Eric, and Brayton stand in line? P(6,6) = 6 x 5 x 4 x 3 x 2 x 1 = 720 different ways There are 6 people to choose from We are selecting this many people (1)(2)(3)(4)(5)(6)

Permutations Example 3: Video Games. If I choose three video games to play at Celebration Station out of ten, in how many different orders can I play those three games?

Permutations Example 3: Video Games. If I choose three video games to play at Celebration Station out of ten, in how many different orders can I play those three games? P(10,3) = 10 x 9 x 8 = 720 different orders We are selecting 3 games to play (1)(2)(3) There are 10 games to choose from

Permutations Example 4: Arrange letters in a word. In how many different ways can you arrange the letters in the word rainbow?

Permutations Example 4: Arrange letters in a word. In how many different ways can you arrange the letters in the word rainbow? P(7,7) = 7 x 6 x 5 x 4 x 3 x 2 x 1 = 5040 ways We are selecting all 7 letters (1)(2)(3) There are 7 different letters to arrange (4)(5)(6)(7)

Permutations Guided Practice: Find the value. (1) P(8,3) = ? (2) How many ways can the three members of the debating team be arranged on the stage?

Permutations Guided Practice: Find the value. (1) P(8,3) = 8 x 7 x 6 = 336 (2) How many ways can the three members of the debating team be arranged on the stage? P(3,3) = 3 x 2 x 1 = 6 ways P(3,3) = 3 x 2 x 1 = 6 ways

Permutations Independent Practice: Find the value. (1) P(6,4) = ? (2) How many ways can 4 books be arranged on a bookshelf?

Permutations Independent Practice: Find the value. (1) P(6,4) = 6 x 5 x 4 x 3 = 360 (2) How many ways can 4 books be arranged on a bookshelf? P(4,4) = 4 x 3 x 2 x 1 = 24 ways P(4,4) = 4 x 3 x 2 x 1 = 24 ways

Permutations Real World Example: Ice Cream. Joselson Ice Cream store has a total of 31 different flavors. They are running a special where you can get three scoops for the price of one. How many ways can you order three different flavored scoops.

Permutations Real World Example: Ice Cream. Joselson Ice Cream store has a total of 31 different flavors. They are running a special where you can get three scoops for the price of one. How many ways can you order three different flavored scoops. P(31,3) = 31 x 30 x 29 = 26,970 different ways Start with this number Count down this many numbers (1)(2)(3)

Permutations Summary: - Permutations involve arrangements or listings where order is important. - We use the following notation: P(9,4) = * The symbol P(9,4) represents the number of permutations of 9 possible things to take, and we are taking 4 of them

Permutations Summary: - Permutations involve arrangements or listings where order is important. - We use the following notation: P(9,4) = 9 x 8 x 7 x 6 = Start with this number Count down this many numbers Permutation

What is Factorial? The Factorial of a specified number refers to the product of a given series of consecutive whole numbers beginning with 1 and ending with the specified number The Factorial of a specified number refers to the product of a given series of consecutive whole numbers beginning with 1 and ending with the specified number We use the “!” to represent factorial We use the “!” to represent factorialEg. 5! = 1  2  3  4  5 = 120 Factorial (not in the range)

It’s A Fact! The number of ways of arranging n objects is n! n! = n  (n − 1)  (n − 2) ...  3  2  1

Calculators 6 x! n! = x!  8 = 9

Using Permutations Factorial Formula for Permutations

Using Permutations How many ways can you select two letters followed by three digits for an ID if repeats are not allowed? Two parts: 2. Determine the set of three digits.1. Determine the set of two letters. 26 P 210 P 3 26   9   ,000

Homework: Pages Permutations