Chapter 12: Counting: Just How Many Are There?

Chapter 12: Counting: Just How Many Are There?

12 Counting Just How Many Are There? Copyright 2014, 2010, 2007 Pearson Education, Inc. Section Section12.2, 1.1, Slide Slide11

12.2 The Fundamental Counting Principle Understand the fundamental counting principle. Use slot diagrams to organize information in counting problems. Know how to solve counting problems with special conditions. Copyright 2014, 2010, 2007 Pearson Education, Inc. Section 12.2, Slide 2

The Fundamental Counting Principle Copyright 2014, 2010, 2007 Pearson Education, Inc. Section 12.2, Slide 3 The Fundamental Counting Principle Example: A group is planning a fund-raising campaign featuring two endangered species (one animal for TV commercials and one for use

online. The list of candidates includes the (C)heetah, the (O)tter, the black-footed (F)erret, and the Bengal (T)iger. In how many ways can we choose the two animals for the campaign? (continued on next slide) Copyright 2014, 2010, 2007 Pearson Education, Inc. Section 12.2, Slide 4 The Fundamental Counting Principle

Solution: We first choose an animal for the TV campaign, which can be done in four ways. We then choose a different animal for the online ads, which can be done in three ways. So the total number of ways to choose the animals is Copyright 2014, 2010, 2007 Pearson Education, Inc. Section 12.2, Slide 5 The Fundamental Counting

Principle Example: How many ways can four coins be flipped? Solution: Copyright 2014, 2010, 2007 Pearson Education, Inc. Section 12.2, Slide 6 The Fundamental Counting Principle

Example: How many ways can four coins be flipped? Solution: Flipping the first coin can be done in two ways. Flipping the second, third, and fourth coins can also each be done in one of two ways. The four coins can be flipped in Copyright 2014, 2010, 2007 Pearson Education, Inc. Section 12.2, Slide 7

The Fundamental Counting Principle Example: How many ways can three dice (red, green, blue) be rolled? Solution: Copyright 2014, 2010, 2007 Pearson Education, Inc. Section 12.2, Slide 8 The Fundamental Counting

Principle Example: How many ways can three dice (red, green, blue) be rolled? Solution: Rolling the red die can be done in six ways. Rolling the green and blue dice can also each be done in one of six ways. The three dice can be rolled in Copyright 2014, 2010, 2007 Pearson Education, Inc. Section 12.2, Slide 9

The Fundamental Counting Principle Example: A summer intern wants to vary his outfit by wearing different combinations of coats, pants, shirts, and ties. If he has three sports coats, five pairs of pants, seven shirts, and four ties, how many different ways can he select an outfit consisting of a coat, pants, shirt, and tie? (continued on next slide) Copyright 2014, 2010, 2007 Pearson Education, Inc.

Section 12.2, Slide 10 The Fundamental Counting Principle Solution: The interns options are Copyright 2014, 2010, 2007 Pearson Education, Inc. Section 12.2, Slide 11 Slot Diagrams

A useful technique for solving problems involving various tasks is to draw a series of blank spaces to keep track of the number of ways to do each task. We will call such a figure a slot diagram. Copyright 2014, 2010, 2007 Pearson Education, Inc. Section 12.2, Slide 12 Slot Diagrams Example: A security keypad uses five digits (0 to 9) in a specific order. How many different

keypad patterns are possible if any digit can be used in any position and repetition is allowed? Solution: Copyright 2014, 2010, 2007 Pearson Education, Inc. Section 12.2, Slide 13 Slot Diagrams Example: A security keypad uses five digits (0 to 9) in a specific order. How many different

keypad patterns are possible if any digit can be used in any position and repetition is allowed? Solution: The slot diagram indicates there are 10 10 10 10 10 = 100,000 possibilities. Copyright 2014, 2010, 2007 Pearson Education, Inc. Section 12.2, Slide 14 Slot Diagrams Example: In the previous example, suppose

the digit 0 cannot be used as the first digit, but otherwise any digit can be used in any position and repetition is allowed. Solution: Copyright 2014, 2010, 2007 Pearson Education, Inc. Section 12.2, Slide 15 Slot Diagrams Example: In the previous example, suppose

the digit 0 cannot be used as the first digit, but otherwise any digit can be used in any position and repetition is allowed. Solution: The slot diagram indicates there are 9 10 10 10 10 = 90,000 possibilities. Copyright 2014, 2010, 2007 Pearson Education, Inc. Section 12.2, Slide 16 Slot Diagrams

Example: In the previous example, suppose any digit can be used in any position, but repetition is not allowed? Solution: Copyright 2014, 2010, 2007 Pearson Education, Inc. Section 12.2, Slide 17 Slot Diagrams Example: In the previous example, suppose any digit can be used in any position, but

repetition is not allowed? Solution: The slot diagram indicates there are 10 9 8 7 6 = 30,240 possibilities. Copyright 2014, 2010, 2007 Pearson Education, Inc. Section 12.2, Slide 18 Handling Special Conditions Example: A college class has 10 students. Louise must sit in the front row next to her tutor. If there are six chairs in the first row of the

classroom, how many different ways can students be assigned to sit in the first row? (continued on next slide) Copyright 2014, 2010, 2007 Pearson Education, Inc. Section 12.2, Slide 19 Handling Special Conditions Example: A college class has 10 students. Louise must sit in the front row next to her tutor. If there are six chairs in the first row of the

classroom, how many different ways can students be assigned to sit in the first row? Solution: We first consider the following tasks: Task 1: Assign two seats to Louise and her tutor. Task 2: Arrange Louise and her tutor in these two seats. Task 3: Assign the remaining seats.(continued on next slide) Copyright 2014, 2010, 2007 Pearson Education, Inc. Section 12.2, Slide 20

Handling Special Conditions Task 1: There are five ways to assign 2 seats to Louise and her tutor. (continued on next slide) Copyright 2014, 2010, 2007 Pearson Education, Inc. Section 12.2, Slide 21 Handling Special Conditions Task 2: There are two ways that Louise and her tutor can sit in their seatsLouise sits either on

the right or the left. Task 3: The remaining four seats are to be filled by four of the eight students left; thus, we have eight students for the first remaining seat, seven for the second seat, and so on. total number possibilities Copyright 2014, 2010, 2007 Pearson Education, Inc. Section 12.2, Slide 22

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