6-7 Permutations & Combinations

1. 6-7 Permutations & Combinations M11.E.3.2.1: Determine the number of permutations and/or combinations or apply the fundamental counting principle

2. Objectives Permutations Combinations

3. Vocabulary A permutation is an arrangement of items in a particular order. n factorial For any positive integer n, n! = n(n – 1) · … · 3 · 2 · 1 For n = 0, n! = 1

4. Vocabulary Fundamental Counting Principle If there are a ways the first event can occur and b ways the second event can occur, then there are a x b ways that both events can occur. Example: You are ordering dinner at a restaurant. You have a choice of soup or salad for an appetizer. A choice of steak, chicken, or tofu for a main entrée, and a choice of pie or ice cream for dessert. How many different meals can you have?

5. Finding Permutations In how many ways can 6 people line up from left to right for a group photo? Since everybody will be in the picture, you are using all the items from the original set. You can use the Multiplication Counting Principle or factorial notation. There are six ways to select the first person in line, five ways to select the next person, and so on. The total number of permutations is 6 • 5 • 4 • 3 • 2 • 1 = 6!. 6! = 720 The 6 people can line up in 720 different orders.

6. Vocabulary Permutations If order does matter, then you are working with permutations. The number of permutations of n items of a set arranged r items at the time is nPr Ex. Seven people are running a race. How many different outcomes of first, second, and third place are possible? n = 7 (total number of items) and r = 3 (how many we’re choosing) This is written 7P3

7. Permutations nPr = for 0 ≤ r ≤ n 10P5= == 5040

8. 26! (26 – 4)! 26! 22! 26P4 = = = 358,800 Real World Example How many 4-letter codes can be made if no letter can be used twice? Method 1: Use the Multiplication Counting Principle. 26 • 25 • 24 • 23 = 358,800 Method 2: Use the permutation formula. Since there are 26 letters arranged 4 at a time, n = 26 and r = 4. There are 358,800 possible arrangements of 4-letter codes with no duplicates.

9. Vocabulary Combinations If order does not matter, then you are working with permutations. The number of combination of n items of a set arranged r items at the time is nCr Ex. Seven people are running a race. How many different ways can 3 people receive a medal? (It doesn’t matter whether they get a gold or bronze) n = 7 (total number of items) and r = 3 (how many we’re choosing) This is written 7C3

10. Combinations nCr = for 0 ≤ r ≤ n 5C3 = == = 10

11. 10! 4!(10 – 4)! 10C4 = 10! 4! • 6! = 2 5 3 4 5 6 6 1 3 2 1 4 10 • 9 • 8 • 7 • 4 • 3 • 2 • 1 • • • • • • = • • • • • 10 • 9 • 8 • 7 4 • 3 • 2 • 1 = Finding Combinations Evaluate 10C4. = 210

12. Relate: 12 songs chosen 5 songs at a time Define: Let n = total number of songs. Let r = number of songs chosen at a time. Write:nCr = Use the nCrfeature of your calculator. 12C5 Real World Example A disk jockey wants to select 5 songs from a new CD that contains 12 songs. How many 5-song selections are possible? You can choose five songs in 792 ways.

13. You may choose  4 toppings, 3 toppings, 2 toppings, 1 toppings, or none. 9C49C39C29C19C0 Real Worl Example A pizza menu allows you to select 4 toppings at no extra charge from a list of 9 possible toppings. In how many ways can you select 4 or fewer toppings? The total number of ways to pick the toppings is 126 + 84 + 36 + 9 + 1 = 256. There are 256 ways to order your pizza.