1. Section 1.1, Slide 1 Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 12.3, Slide 1 12 Counting Just How Many Are There?

2. Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 12.3, Slide 2 Permutations and Combinations 12.3 Calculate the number of permutations of n objects taken r at a time. Use factorial notation to represent the number of permutations of a set of objects.

3. Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 12.3, Slide 3 Permutations and Combinations 12.3 Calculate the number of combinations of n objects taken r at a time. Apply the theory of permutations and combinations to solve counting problems.

4. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.3, Slide 4 Permutations

5. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.3, Slide 5 Example: How many permutations are there of the letters a, b, c, and d ? Write the answer using P( n, r ) notation. Solution: We write the letters a, b, c, and d in a line without repetition, so abcd and bcad are two such permutations. Permutations (continued on next slide)

6. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.3, Slide 6 The slot diagram indicates there are 4 × 3 × 2 × 1 = 24 possibilities. This can be written more succinctly as Permutations

7. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.3, Slide 7 Example: How many permutations are there of the letters a, b, c, d, e, f, and g if we take the letters three at a time? Write the answer using P(n, r) notation. Permutations Solution: The slot diagram indicates there are 7 × 6 × 5 = 210 possibilities. This can be written in permutation notation as shown.

8. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.3, Slide 8 Factorial Notation

9. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.3, Slide 9 Example: Compute (8 – 3)!. Solution: We work inside parentheses first. Factorial Notation

10. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.3, Slide 10 Example: Compute. Solution: Factorial Notation

11. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.3, Slide 11 Factorial Notation

12. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.3, Slide 12 Example: The 12-person theater group wishes to select one person to direct a play, a second to supervise the music, and a third to handle publicity, tickets, and other administrative details. In how many ways can the group fill these positions? Solution: This is a permutation of selecting 3 people from 12. Factorial Notation

13. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.3, Slide 13 Combinations

14. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.3, Slide 14 Example: How many three-element sets can be chosen from a set of five objects? Solution: Order is not important, so it is clear that this is a combination problem. Combinations

15. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.3, Slide 15 Example: How many four-person committees can be formed from a set of 10 people? Solution: Order is not important, so it is clear that this is a combination problem. Combinations

16. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.3, Slide 16 Example: A syndicate has $15 million to spend on tickets for a lottery. Tickets cost $1 and contain a combination of six numbers from 1 to 44. Does the syndicate have enough money to buy enough tickets to be guaranteed a winner? Solution : The combination of 6 numbers from the 44 possible gives the number of different tickets. That is, C(44, 6) = 7,059,052. Thus, the syndicate has more than enough money to buy enough tickets to be guaranteed a winner. Combinations

17. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.3, Slide 17 Example: In the game of poker, five cards are drawn from a standard 52-card deck. How many different poker hands are possible? Solution : Combinations

18. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.3, Slide 18 Example: In the game of bridge, a hand consists of 13 cards drawn from a standard 52- card deck. How many different bridge hands are there? Solution : Combinations

19. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.3, Slide 19 Example: Two men and two women from a firm will attend a conference. If the firm has ten men and nine women, in how many different ways can the conference attendees be selected? Solution : The answer is not C(19, 4) since this includes options like four men and no woman being sent to the conference. Combining Counting Methods (continued on next slide)

20. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.3, Slide 20 Stage 1: Select the two women from the nine available. Stage 2: Select the two men from the ten available. Thus, choosing the women and then choosing the men can be done in ways. Combining Counting Methods

21. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.3, Slide 21 Example: A 16-member consortium wishes to choose a committee consisting of a president, a vice president, and a three-member executive board. In how many different ways can this committee be formed? Solution : We will count this in two stages: (a) choosing the president and vice president from the consortium, (b) choosing an executive board from the remaining members. Combining Counting Methods (continued on next slide)

22. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.3, Slide 22 Stage 1: Choose the president and vice president. This can be done in P(16, 2) ways. Stage 2: Select the executive board. This can be done in C(14, 3) ways. Total: Combining Counting Methods

23. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.3, Slide 23 Combining Counting Methods

24. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.3, Slide 24 Combining Counting Methods For example, consider the set {1, 2, 3, 4} and the 4 th row of Pascal’s triangle: 1 4 6 4 1.

25. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.3, Slide 25 Combining Counting Methods For example, consider the 4 th row of Pascal’s triangle: 1 4 6 4 1. C(4, 0) = 1 C(4, 1) = 4 C(4, 2) = 6 C(4, 3) = 4 C(4, 4) = 1

26. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.3, Slide 26 Example: Assume that a pharmaceutical company has developed five antibiotics and four immune system stimulators. In how many ways can we choose a treatment program consisting of three antibiotics and two immune system stimulators to treat a disease? Use Pascal’s triangle to speed your computations. Solution : We will count this in two stages: (a) choosing the antibiotics, (b) choosing the immune system simulators. Combining Counting Methods (continued on next slide)

27. Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 12.3, Slide 27 Stage 1: Choosing 3 antibiotics from 5 can be done in C(5, 3) ways. Stage 2: Choosing 2 immune system simulators from 4 can be done in C(4, 2) ways. Total: C(5, 3) × C(4, 2) = 10 × 6 = 60 ways. Combining Counting Methods