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Chapter 10

Chapter 10. Counting Methods. Chapter 10: Counting Methods. 10.1 Counting by Systematic Listing 10.2 Using the Fundamental Counting Principle 10.3 Using Permutations and Combinations 10.4 Using Pascal’s Triangle 10.5 Counting Problems Involving “Not” and “Or”. Section 10-1.

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Chapter 10

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  1. Chapter 10 Counting Methods 2012 Pearson Education, Inc.

  2. Chapter 10: Counting Methods • 10.1 Counting by Systematic Listing • 10.2 Using the Fundamental Counting Principle • 10.3 Using Permutations and Combinations • 10.4 Using Pascal’s Triangle • 10.5 Counting Problems Involving “Not” and “Or” 2012 Pearson Education, Inc.

  3. Section 10-1 • Counting by Systematic Listing 2012 Pearson Education, Inc.

  4. Counting by Systematic Listing • One-Part Tasks • Product Tables for Two-Part Tasks • Tree Diagrams for Multiple-Part Tasks • Other Systematic Listing Methods 2012 Pearson Education, Inc.

  5. One-Part Tasks The results for simple, one-part tasks can often be listed easily. For the task of tossing a fair coin, the list is heads, tails, with two possible results. For the task of rolling a single fair die the list is 1, 2, 3, 4, 5, 6, with six possibilities. 2012 Pearson Education, Inc.

  6. Example: Selecting a Club President Consider a club N with four members: N = {Mike, Adam, Ted, Helen} or in abbreviated form N = {M, A, T, H} In how many ways can this group select a president? Solution The task is to select one of the four members as president. There are four possible results: M, A, T, and H. 2012 Pearson Education, Inc.

  7. Example: Product Tables for Two-Part Tasks Determine the number of two-digit numbers that can be written using the digits from the set {2, 4, 6}. Solution The task consists of two parts: 1. Choose a first digit 2. Choose a second digit The results for a two-part task can be pictured in a product table, as shown on the next slide. 2012 Pearson Education, Inc.

  8. Example: Product Tables for Two-Part Tasks Solution(continued) From the table we obtain the list of possible results: 22, 24, 26, 42, 44, 46, 62, 64, 66. 2012 Pearson Education, Inc.

  9. Example: Possibilities for Rolling a Pair of Distinguishable Dice 2012 Pearson Education, Inc.

  10. Example: Electing Club Officers Find the number of ways club N (previous slide) can elect a president and secretary. Solution The task consists of two parts: 1. Choose a president 2. Choose a secretary The product table is pictured on the next slide. 2012 Pearson Education, Inc.

  11. Example: Product Tables for Two-Part Tasks Solution(continued) From the table we see that there are 12 possibilities. 2012 Pearson Education, Inc.

  12. Example: Electing Club Officers Find the number of ways club N (previous slide) can appoint a committee of two members. Solution Using the table on the previous slide, this time the order of the letters (people) in a pair makes no difference. So there are 6 possibilities: MA, MT, MH, AT, AH, TH. 2012 Pearson Education, Inc.

  13. Tree Diagrams for Multiple-Part Tasks A task that has more than two parts is not easy to analyze with a product table. Another helpful device is a tree diagram, as seen in the next example. 2012 Pearson Education, Inc.

  14. Example: Building Numbers From a Set of Digits Find the number of three digit numbers that can be written using the digits from the set {2, 4, 6} assuming repeated digits are not allowed. Solution Second Third First 4 6 6 4 246 264 426 462 624 642 2 4 6 2 6 6 2 6 possibilities 2 4 4 2 2012 Pearson Education, Inc.

  15. Other Systematic Listing Methods There are additional systematic ways to produce complete listings of possible results besides product tables and tree diagrams. One of these ways is shown in the next example. 2012 Pearson Education, Inc.

  16. Example: Counting Triangles in a Figure How many triangles (of any size) are in the figure below? D Solution E C One systematic approach is to label the points as shown, begin with A, and proceed in alphabetical F A B order to write all 3-letter combinations (like ABC, ABD, …), then cross out ones that are not triangles. There are 12 different triangles. 2012 Pearson Education, Inc.

  17. Section 10-2 • Using the Fundamental Counting Principle 2012 Pearson Education, Inc.

  18. Using the Fundamental Counting Principle • Uniformity and the Fundamental Counting Principle • Factorials • Arrangements of Objects 2012 Pearson Education, Inc.

  19. Uniformity Criterion for Multiple-Part Tasks A multiple-part task is said to satisfy the uniformity criterion if the number of choices for any particular part is the same no matter which choices were selected for the previous parts. 2012 Pearson Education, Inc.

  20. Fundamental Counting Principle When a task consists of k separate parts and satisfies the uniformity criterion, if the first part can be done in n1 ways, the second part can be done in n2 ways, and so on through the k th part, which can be done in nk ways, then the total number of ways to complete the task is given by the product 2012 Pearson Education, Inc.

  21. Example: Two-Digit Numbers How many two-digit numbers can be made from the set {0, 1, 2, 3, 4, 5}? (numbers can’t start with 0.) Solution There are 5(6) = 30 two-digit numbers. 2012 Pearson Education, Inc.

  22. Example: Two-Digit Numbers with Restrictions How many two-digit numbers that do not contain repeated digits can be made from the set {0, 1, 2, 3, 4, 5} ? Solution There are 5(5) = 25 two-digit numbers. 2012 Pearson Education, Inc.

  23. Example: Two-Digit Numbers with Restrictions How many ways can you select two letters followed by three digits for an ID? Solution There are 26(26)(10)(10)(10) = 676,000 IDs possible. 2012 Pearson Education, Inc.

  24. Factorials For any counting number n, the product of all counting numbers from n down through 1 is called n factorial, and is denoted n!. 2012 Pearson Education, Inc.

  25. Factorial Formula For any counting number n, the quantity n factorial is given by 2012 Pearson Education, Inc.

  26. Example: Evaluate each expression. a) 4! b) (4 – 1)! c) Solution 2012 Pearson Education, Inc.

  27. Definition of Zero Factorial 2012 Pearson Education, Inc.

  28. Arrangements of Objects When finding the total number of ways to arrange a given number of distinct objects, we can use a factorial. 2012 Pearson Education, Inc.

  29. Arrangements of n Distinct Objects The total number of different ways to arrange n distinct objects is n!. 2012 Pearson Education, Inc.

  30. Example: Arranging Books How many ways can you line up 6 different books on a shelf? Solution The number of ways to arrange 6 distinct objects is 6! = 720. 2012 Pearson Education, Inc.

  31. Arrangements of n Objects Containing Look-Alikes The number of distinguishable arrangements of n objects, where one or more subsets consist of look-alikes (say n1 are of one kind, n2 are of another kind, …, and nk are of yet another kind), is given by 2012 Pearson Education, Inc.

  32. Example: Distinguishable Arrangements Determine the number of distinguishable arrangements of the letters of the word INITIALLY. Solution 9 letters total 3 I’s and 2 L’s 2012 Pearson Education, Inc.

  33. Section 10-3 • Using Permutations and Combinations 2012 Pearson Education, Inc.

  34. Using Permutations and Combinations • Permutations • Combinations • Guidelines on Which Method to Use 2012 Pearson Education, Inc.

  35. Permutations In the context of counting problems, arrangements are often called permutations; the number of permutations of n things taken r at a time is denoted nPr. Applying the fundamental counting principle to arrangements of this type gives nPr= n(n – 1)(n – 2)…[n – (r – 1)]. 2012 Pearson Education, Inc.

  36. Factorial Formula for Permutations The number of permutations, or arrangements, of n distinct things taken r at a time, where rn, can be calculated as 2012 Pearson Education, Inc.

  37. Example: Permutations Evaluate each permutation. a) 5P3 b) 6P6 Solution 2012 Pearson Education, Inc.

  38. Example: IDs How many ways can you select two letters followed by three digits for an ID if repeats are not allowed? Solution There are two parts: 1. Determine the set of two letters. 2. Determine the set of three digits. Part 1 Part 2 2012 Pearson Education, Inc.

  39. Example: Building Numbers From a Set of Digits How many four-digit numbers can be written using the numbers from the set {1, 3, 5, 7, 9} if repetitions are not allowed? Solution Repetitions are not allowed and order is important, so we use permutations: 2012 Pearson Education, Inc.

  40. Combinations In the context of counting problems, subsets, where order of elements makes no difference, are often called combinations; the number of combinations of n things taken r at a time is denoted nCr. 2012 Pearson Education, Inc.

  41. Factorial Formula for Combinations The number of combinations, or subsets, of n distinct things taken r at a time, where rn, can be calculated as Note: 2012 Pearson Education, Inc.

  42. Example: Combinations Evaluate each combination. a) 5C3 b) 6C6 Solution 2012 Pearson Education, Inc.

  43. Example: Finding the Number of Subsets Find the number of different subsets of size 3 in the set {m, a, t, h, r, o, c, k, s}. Solution A subset of size 3 must have 3 distinct elements, so repetitions are not allowed. Order is not important. 2012 Pearson Education, Inc.

  44. Example: Finding the Number of Subsets A common form of poker involves hands (sets) of five cards each, dealt from a deck consisting of 52 different cards. How many different 5-card hands are possible? Solution Repetitions are not allowed and order is not important. 2012 Pearson Education, Inc.

  45. Guidelines on Which Method to Use 2012 Pearson Education, Inc.

  46. Section 10-4 • Using Pascal’s Triangle 2012 Pearson Education, Inc.

  47. Using Pascal’s Triangle • Pascal’s Triangle • Applications 2012 Pearson Education, Inc.

  48. Pascal’s Triangle The triangular array on the next slide represents “random walks” that begin at START and proceed downward according to the following rule. At each circle (branch point), a coin is tossed. If it lands heads, we go downward to the left. If it lands tails, we go downward to the right. At each point, left an right are equally likely. In each circle the number of different routes that could bring us to that point are recorded. 2012 Pearson Education, Inc.

  49. Pascal’s Triangle START 1 1 1 1 2 1 1 1 3 3 1 4 1 4 6 2012 Pearson Education, Inc.

  50. Pascal’s Triangle Another way to generate the same pattern of numbers is to begin with 1s down both diagonals and then fill in the interior entries by adding the two numbers just above the given position. The pattern is shown on the next slide. This unending “triangular array of numbers is called Pascal’s triangle. 2012 Pearson Education, Inc.

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