Copyright © 2014, 2010, 2007 Pearson Education, Inc. Chapter 8 Sequences, Induction, and Probability 8.6 Counting Principles, Permutations, and Combinations Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1

Objectives: Use the Fundamental Counting Principle. Use the permutations formula. Distinguish between permutation problems and combination problems. Use the combinations formula.

The Fundamental Counting Principle The number of ways in which a series of successive things can occur is found by multiplying the number of ways in which each thing can occur.

Example: Using the Fundamental Counting Principle A pizza can be ordered with three choices of size (small, medium, or large), four choices of crust (thin, thick, crispy, or regular), and six choices of toppings (ground beef, sausage, pepperoni, bacon, mushrooms, or onions). How many different one-topping pizzas can be ordered? We use the Fundamental Counting Principle to find the number of different one-topping pizzas that can be ordered. Size – 3 choices Crust – 4 choices Toppings – 6 choices 72 different one-topping pizzas can be ordered.

Permutations A permutation is an ordered arrangement of items that occurs when No item is used more than once. The order of arrangement makes a difference. The number of possible permutations if r items are taken from n items is

Example: Using the Formula for Permutations In how many ways can six books be lined up along a shelf? Because we are using all six books in every possible arrangement, we are arranging r = 6 books from a group of n = 6 books. Six books can be lined up along a shelf in 720 ways.

Combinations A combination of items occurs when The items are selected from the same group. No item is used more than once. The order of the items makes no difference.

Example: Distinguishing between Permutations and Combinations For each of the following problems, explain if the problem is one involving permutations or combinations. a. How many ways can you select 6 free DVD’s from a list of 200 DVD’s? Because order makes no difference, this is a combination. b. In a race in which there are 50 runners and no ties, in how many ways can the first three finishers come in? The order in which each runner finishes makes a difference, this is a permutation.

Combinations of n Things Taken r at a Time The number of possible combinations if r items are taken from n items is

Example: Using the Formula for Combinations From a group of 10 physicians, in how many ways can four people be selected to attend a conference on acupuncture? We are selecting r = 4 people from a group of n = 10 people. Four people can be selected from a group of 10 in 210 ways.