1. Section 1.1, Slide 1 Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 12.1, Slide 1 12 Counting Just How Many Are There?

2. Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 12.1, Slide 2 Introduction to Counting Methods 12.1 Count elements in a set systematically. Use tree diagrams to represent counting situations graphically. Use counting techniques to solve applied problems.

3. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.1, Slide 3 Systematic Counting Example: How many ways can we do each of the following? (continued on next slide) a) Flip a coin. b) Roll a single die (singular of dice). c) Pick a card from a standard deck of cards. d) Choose a features editor from a five- person newspaper staff.

4. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.1, Slide 4 Solution: a)The coin can come up either heads or tails, so there are two ways to flip a coin. b) The die has six faces numbered 1, 2, 3, 4, 5, and 6, so there are six ways the die can be rolled. c) There are 52 different ways to choose a card from a standard deck. d) There are five ways to choose one of the five staff members to be editor. Systematic Counting

5. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.1, Slide 5 Systematic Counting 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)

6. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.1, Slide 6 Systematic Counting Solution: We begin by assuming that C is the animal selected for the TV commercials and then consider each of the other animals for the online campaign to get CO, CF, CT. doing this for all the animals, we get CO, CF, CT OC, OF, OT FC, FO, FT TC, TO, TF. That is, we have 12 different ways to choose the two animals.

7. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.1, Slide 7 Tree Diagrams Example: How many ways can three coins be flipped? Solution: Let’s assume we are flipping a penny, a nickel, and a dime. A tree diagram is a handy way to illustrate the possibilities. (continued on next slide) First illustrate the possibilities for flipping the penny.

8. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.1, Slide 8 Tree Diagrams (continued on next slide) Next illustrate the possibilities for the nickel after having already flipped the penny.

9. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.1, Slide 9 Tree Diagrams Now illustrate the possibilities for the dime after having already flipped the penny and the nickel. We can trace eight branches that indicate the ways the three coins can be flipped:

10. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.1, Slide 10 Tree Diagrams Example: If we roll two dice, how many different pairs of numbers can appear on the upturned faces? Solution: (continued on next slide)

11. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.1, Slide 11 Tree Diagrams Example: If we roll two dice, how many different pairs of numbers can appear on the upturned faces? Solution: We will use ordered pairs of the form (red number, green number) to represent the pairs showing on the dice. We can roll a 1 on the red die and either a 1, 2, 3, 4, 5, or 6 on the green. This corresponds to the pairs (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), and (1, 6). (continued on next slide)

12. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.1, Slide 12 Tree Diagrams We use a tree diagram to illustrate the possibilities. We see that there are 36 different possibilities.

13. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.1, Slide 13 Tree Diagrams Example: The director at a local TV station wants to fill three commercial spots using promos for the latest albums by singers (J)ordin, (T)aylor, and (C)arrie. In how many ways can these spots be filled if repetition is allowed? Solution : (continued on next slide)

14. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.1, Slide 14 Tree Diagrams Example: The director at a local TV station wants to fill three commercial spots using promos for the latest albums by singers (J)ordin, (T)aylor, and (C)arrie. In how many ways can these spots be filled if repetition is allowed? Solution : If repetition is allowed, then we could choose J, T, and then T again. We will abbreviate this ordering as JTT. (continued on next slide)

15. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.1, Slide 15 Tree Diagrams The tree diagram illustrates all possibilities. We see that there are 27 ways to fill the promo slots.

16. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.1, Slide 16 Tree Diagrams Example: In how many ways can the promo spots be filled in the previous example if repetition is not allowed? Solution : If repetition is not allowed, then we are not allowed to choose JTT. (continued on next slide)

17. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.1, Slide 17 Tree Diagrams The tree diagram illustrates all possibilities. We see that there are 6 ways to fill the promo slots.

18. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.1, Slide 18 Visualizing Trees Example: A designer has created five tops, four pairs of pants, and three jackets. If we consider an outfit to be a top, pants, and jacket, how many different outfits can be formed without repeating the exact same outfit? Solution :

19. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.1, Slide 19 Visualizing Trees Example: A designer has created five tops, four pairs of pants, and three jackets. If we consider an outfit to be a top, pants, and jacket, how many different outfits can be formed without repeating the exact same outfit? Solution : For a tree diagram, we would start with 5 branches (tops), then we would attach 4 branches to each top (pants) giving 20 branches. We would then attach 3 branches (jackets). This gives a total of 60 different branches (outfits).