Counting

1. Counting Just How Many Are There?

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. Systematic Counting • Example: How many ways can we do each of the following? 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. (continued on next slide)

4. Systematic Counting • Solution: • 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.

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. 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. Systematic Counting • Tree Diagram Illustration:

8. Tree Diagrams • Example: How many ways can three coins be flipped? (continued on next slide)

9. 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 diagramis a handy way to illustrate the possibilities. First illustrate the possibilities for flipping the penny. (continued on next slide)

10. Tree Diagrams Next illustrate the possibilities for the nickel after having already flipped the penny. (continued on next slide)

11. 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:

12. Tree Diagrams • Example: If we roll two dice, how many different pairs of numbers can appear on the upturned faces? (continued on next slide)

13. 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)

14. Tree Diagrams We use a tree diagramto illustrate the possibilities. We see that there are 36 different possibilities.

15. 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? (continued on next slide)

16. 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)

17. Tree Diagrams The tree diagram illustrates all possibilities. We see that there are 27 ways to fill the promo slots.

18. Tree Diagrams • Example: In how many ways can the promo spots be filled in the previous example if repetition is not allowed? (continued on next slide)

19. 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)

20. Tree Diagrams The tree diagram illustrates all possibilities. We see that there are 6 ways to fill the promo slots.

21. 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?

22. 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).