Chapter 7: Counting Principles

1. Chapter 7: Counting Principles Discrete Mathematical Structures: Theory and Applications

2. Learning Objectives • Learn the basic counting principles—multiplication and addition • Explore the pigeonhole principle • Learn about permutations • Learn about combinations Discrete Mathematical Structures: Theory and Applications

3. Learning Objectives • Explore generalized permutations and combinations • Learn about binomial coefficients and explore the algorithm to compute them • Discover the algorithms to generate permutations and combinations • Become familiar with discrete probability Discrete Mathematical Structures: Theory and Applications

4. Basic Counting Principles Discrete Mathematical Structures: Theory and Applications

5. Basic Counting Principles Discrete Mathematical Structures: Theory and Applications

6. Basic Counting Principles • There are three boxes containing books. The first box contains 15 mathematics books by different authors, the second box contains 12 chemistry books by different authors, and the third box contains 10 computer science books by different authors. • A student wants to take a book from one of the three boxes. In how many ways can the student do this? Discrete Mathematical Structures: Theory and Applications

7. Basic Counting Principles • Suppose tasks T1, T2, and T3 are as follows: • T1: Choose a mathematics book. • T2: Choose a chemistry book. • T3: Choose a computer science book. • Then tasks T1, T2, and T3 can be done in 15, 12, and 10 ways, respectively. • All of these tasks are independent of each other. Hence, the number of ways to do one of these tasks is 15 + 12 + 10 = 37. Discrete Mathematical Structures: Theory and Applications

8. Basic Counting Principles Discrete Mathematical Structures: Theory and Applications

9. Basic Counting Principles • Morgan is a lead actor in a new movie. She needs to shoot a scene in the morning in studio A and an afternoon scene in studio C. She looks at the map and finds that there is no direct route from studio A to studio C. Studio B is located between studios A and C. Morgan’s friends Brad and Jennifer are shooting a movie in studio B. There are three roads, say A1, A2, and A3, from studio A to studio B and four roads, say B1, B2, B3, and B4, from studio B to studio C. In how many ways can Morgan go from studio A to studio C and have lunch with Brad and Jennifer at Studio B? Discrete Mathematical Structures: Theory and Applications

10. Basic Counting Principles • There are 3 ways to go from studio A to studio B and 4 ways to go from studio B to studio C. • The number of ways to go from studio A to studio C via studio B is 3 * 4 = 12. Discrete Mathematical Structures: Theory and Applications

11. Basic Counting Principles Discrete Mathematical Structures: Theory and Applications

12. Basic Counting Principles • Consider two finite sets, X1and X2. Then • This is called the inclusion-exclusion principle for two finite sets. • Consider three finite sets, A, B, and C. Then • This is called the inclusion-exclusion principle for three finite sets. Discrete Mathematical Structures: Theory and Applications

13. Basic Counting Principles Discrete Mathematical Structures: Theory and Applications

14. Pigeonhole Principle • The pigeonhole principle is also known as the Dirichlet drawer principle, or the shoebox principle. Discrete Mathematical Structures: Theory and Applications

15. Pigeonhole Principle Discrete Mathematical Structures: Theory and Applications

16. Discrete Mathematical Structures: Theory and Applications

17. Pigeonhole Principle Discrete Mathematical Structures: Theory and Applications

18. Permutations Discrete Mathematical Structures: Theory and Applications

19. Permutations Discrete Mathematical Structures: Theory and Applications

20. Combinations Discrete Mathematical Structures: Theory and Applications

21. Combinations Discrete Mathematical Structures: Theory and Applications

22. Generalized Permutations and Combinations Discrete Mathematical Structures: Theory and Applications

23. Generalized Permutations and Combinations Discrete Mathematical Structures: Theory and Applications

24. Binomial Coefficients • The expression x +y isa binomial expression as it is the sum of two terms. • The expression (x +y)nis called a binomial expression of order n. Discrete Mathematical Structures: Theory and Applications

25. Binomial Coefficients Discrete Mathematical Structures: Theory and Applications

26. Binomial Coefficients Discrete Mathematical Structures: Theory and Applications

27. Binomial Coefficients • Pascal’s Triangle • The number C(n, r) can be obtainedby constructing a triangular array. • The row 0, i.e., the first row of the triangle, contains the single entry 1. The row 1, i.e., the second row, contains a pair of entries each equal to 1. • Calculate the nth row of the triangle from the preceding row by the following rules: Discrete Mathematical Structures: Theory and Applications

28. Binomial Coefficients Discrete Mathematical Structures: Theory and Applications

29. Discrete Mathematical Structures: Theory and Applications

30. Binomial Coefficients • ALGORITHM 7.1: Determine the factorial of a nonnegative integer. • Input: n—a positive integer • Output: n! • function factorial(n) • begin • fact := 1; • for i := 2 to n do • fact := fact * i; • return fact; • end Discrete Mathematical Structures: Theory and Applications

31. Binomial Coefficients • The technique known as divide and conquer can be usedto compute C(n, r ). • In the divide-and-conquer technique, a problem is divided into a fixed number, say k, of smaller problems of the same kind. • Typically, k = 2. Each of the smaller problems is then divided into k smaller problems of the same kind, and so on, until the smaller problem is reduced to a case in which the solution is easily obtained. • The solutions of the smaller problems are then put together to obtain the solution of the original problem. Discrete Mathematical Structures: Theory and Applications

32. Binomial Coefficients Discrete Mathematical Structures: Theory and Applications

33. Binomial Coefficients • ALGORITHM 7.3: Determine C(n, r) using dynamic programming. • Input: n, r , n > 0, r > 0, r ≤ n • Output: C(n, r) • function combDynamicProg(n,r) • begin • for i := 0 to n do • for j := 0 to min(i,r) do • if j = 0 or j = i then • C[i,j] := 1; • else • C[i,j] := C[i-1, j-1] + C[i-1, j]; • return C[n, r]; • end Discrete Mathematical Structures: Theory and Applications

34. Generating Permutations and Combinations Discrete Mathematical Structures: Theory and Applications

35. Generating Permutations and Combinations Discrete Mathematical Structures: Theory and Applications

36. Generating Permutations and Combinations Discrete Mathematical Structures: Theory and Applications

37. Generating Permutations and Combinations Discrete Mathematical Structures: Theory and Applications

38. Generating Permutations and Combinations Discrete Mathematical Structures: Theory and Applications

39. Generating Permutations and Combinations Discrete Mathematical Structures: Theory and Applications

40. Discrete Mathematical Structures: Theory and Applications

41. Discrete Mathematical Structures: Theory and Applications

42. Discrete Probability • Definition 7.8.1 • A probabilistic experiment, or random experiment, or simply an experiment, is the process by which an observation is made. • In probability theory, any action or process that leads to an observation is referred to as an experiment. • Examples include: • Tossing a pair of fair coins. • Throwing a balanced die. • Counting cars that drive past a toll booth. Discrete Mathematical Structures: Theory and Applications

43. Discrete Probability • Definition 7.8.3 • The sample space associated with a probabilistic experiment is the set consisting of all possible outcomes of the experiment and is denoted by S. • The elements of the sample space are referred to as sample points. • A discrete sample space is one that contains either a finite or a countable number of distinct sample points. Discrete Mathematical Structures: Theory and Applications

44. Discrete Probability • Definition 7.8.6 • An event in a discrete sample space S is a collection of sample points, i.e., any subset of S. In other words, an event is a set consisting of possible outcomes of the experiment. • Definition 7.8.7 • A simple event is an event that cannot be decomposed. Each simple event corresponds to one and only one sample point. Any event that can be decomposed into more than one simple event is called a compound event. Discrete Mathematical Structures: Theory and Applications

45. Discrete Probability • Definition 7.8.8 • Let A be an event connected with a probabilistic experiment E and let S be the sample space of E. The event B of nonoccurrence of A is called the complementary event of A. This means that the subset B is the complement A’of A in S. • In an experiment, two or more events are said to be equally likely if, after taking into consideration all relevant evidences, none can be expected in reference to another. Discrete Mathematical Structures: Theory and Applications

46. Discrete Probability Discrete Mathematical Structures: Theory and Applications

47. Discrete Probability • Axiomatic Approach • Analyzing the concept of equally likely probability, we see that three conditions must hold. • The probability of occurrence of any event must be greater than or equal to 0. • The probability of the whole sample space must be 1. • If two events are mutually exclusive, the probability of their union is the sum of their respective probabilities. • These three fundamental concepts form the basis of the definition of probability. Discrete Mathematical Structures: Theory and Applications

48. Discrete Probability Discrete Mathematical Structures: Theory and Applications

49. Discrete Probability Discrete Mathematical Structures: Theory and Applications

50. Discrete Probability Discrete Mathematical Structures: Theory and Applications