Report

Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. SECTION 4.4 COUNTING Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Objectives 1. 2. 3. Count the number of ways a sequence of operations can be performed Count the number of permutations Count the number of combinations Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Objective 1 Count the number of ways a sequence of operations can be performed Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. The Fundamental Principle of Counting If an operation can be performed in m ways, and a second operation can be performed in n ways, then the total number of ways to perform the sequence of two operations is mn. If a sequence of several operations is to be performed, the number of ways to perform the sequence is found by multiplying together the numbers of ways to perform each of the operations. Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example A certain make of automobile is available in any of three colors: red, blue, or green, and comes with either a large or small engine. In how many ways can a buyer choose a car? Solution: There are 3 choices of color and 2 choices of engine. The total number of choices is 3·2 = 6 Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example License plates in a certain state contain three letters followed by three digits. How many different license plates can be made? Solution: There are six operations in all; choosing three letters and choosing three digits. There are 26 ways to choose each letter and 10 ways to choose each digit. The total number of license plates is therefore 26·26·26·10·10·10 = 17,576,000. Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Objective 2 Count the number of permutations Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Definition: Permutation The word “permutation” is another word for “ordering.” When we count the number of permutations, we are counting the number of different ways that a group of items can be ordered. Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Permutation of n Objects The number of permutations of n objects is n! Definition of n! For any positive integer n, the number n! is pronounced “n factorial” and is equal to the product of all the integers from n down to 1. n! = n(n − 1)(n –2)∙ ∙ ∙ 3(2)(1) By definition, 0! = 1. Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example Five runners run a race. One of them will finish first, another will finish second, and so on. In how many different orders can they finish? Solution: The number of different orders in which the runners can finish is 5! = 5·4·3·2·1 = 120. Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example Ten runners run a race. The first-place finisher will win a gold medal, the second-place finisher will win a silver medal, and the third-place finisher will win a bronze medal. In how many different ways can the medals be awarded? Solution: We use the Fundamental Principle of Counting. There are 10 possible choices for the gold medal winner. Once the gold medal winner is determined, there are nine remaining choices for the silver medal. Finally, there are eight choices for the bronze medal. The total number of ways the medals can be awarded is10·9·8 = 720. Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Permutation of r Objects Chosen from n In the last example, three runners were chosen from a group of ten, then ordered as first, second, and third. This is referred to as a permutation of three items chosen from ten. In general, a permutation of r items chosen from n items is an ordering of the r items. It is obtained by choosing r items from a group of n items, then choosing an order for the r items. The number of permutations of r items chosen from n is denoted nPr. The number of permutations of r objects chosen from n is n! P n ( n 1) ( n r 1) n r (n r )! Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example Five lifeguards are available for duty one Saturday afternoon. There are three lifeguard stations. In how many ways can three lifeguards be chosen and ordered among the stations? Solution: We are choosing three items from a group of five and ordering them. The number of ways to do this is 5 P3 5! 5! 5 4 3 2 1 5 4 3 60 (5 3)! 2! 2 1 Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Objective 3 Count the number of combinations Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Combinations In some cases, when choosing a set of objects from a larger set, we don’t care about the ordering of the chosen objects; we care only which objects are chosen. For example, we may not care which lifeguard occupies which station; we might care only which three lifeguards are chosen. Each distinct group of objects that can be selected, without regard to order, is called a combination. The number of combinations of r objects chosen from a group of n objects is n Cr n! r !(n r )! Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example At a certain event, 30 people attend, and 5 will be chosen at random to receive prizes. The prizes are all the same, so the order in which the people are chosen does not matter. How many different groups of 5 people can be chosen? Solution: Since the order of the 5 chosen people does not matter, we need to compute the number of combinations of 5 chosen from 30. This is 30 C5 30! 30 29 28 27 26 142,506 5!(30 5)! 5 4 3 2 1 Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Permutations and Combinations on the TI-84 PLUS Calculator commands for permutations and combinations are accessed by pressing MATH and scrolling to the PRB menu. The format for entering these commands is to enter the value of n into the calculator, select either nPr (permutations) or nCr (combinations) from the PRB menu, and then enter the value of r. To run the command, press Enter. Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Do You Know • • • How to use the Fundamental Principle of Counting to count the number of ways a sequence of operations can be performed? How to count the number of permutations? How to count the number of combinations? Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.