Part II: Counting

1. Part II: Counting • Basic Counting: Principles • Lists, Permutations, and Subsets • Binomial Coefficients • Pigeonhole Principle

2. L04: Basic Counting • Objectives • Counting: What and Why? • Basic principles of counting • Reading • SDB, pp. 31-38

3. Outline • Introduction to Counting • The Sum Principle • Principle through an example • Set-Theoretic concepts and Notations • Summing consecutive integers • The Product Principle • The principle • An application: Two-Elements subsets

8. Outline • Introduction to Counting • The Sum Principle • Principle through an example • Set-Theoretic concepts and Notations • Summing consecutive integers • The Product Principle • The principle • An application: Two-Elements subsets

10. Selection Sort Comparison Counting # of comparisons: n n n 2 2 2 n-1 n-1 n-1 1 1 1 3 3 3 … … … … … … i=1: n-1 i=2: n-2 ….. i=n-1: 1 Total # of comparisons = (n-1)+(n-2)+…+1

11. Sum Principle S # of comparisons: =|S1| n n n 2 2 2 n-1 n-1 n-1 1 1 1 3 3 3 … … … … … … i=1: n-1 i=2: n-2 ….. i=n-1: 1 Total # of comparisons = (n-1)+(n-2)+…+1 =|S2| =|Sn-1|

12. Sum Principle in Abstract S4 S2 S1 S3 Si: blocks subsets … … Abstraction: From Concrete example to General Principle

13. Set-Theoretical Concepts in Sum Principle

14. Outline • Introduction to Counting • The Sum Principle • Principle through an example • Set-Theoretic concepts and Notations • Summing consecutive integers • The Product Principle • The principle • An application: Two-Elements subsets

15. {p|p an UST student who takes 91M to school}

21. Outline • Introduction to Counting • The Sum Principle • Principle through an example • Set-Theoretic concepts and Notations • Summing consecutive integers • The Product Principle • The principle • An application: Two-Elements subsets

24. Outline • Introduction to Counting • The Sum Principle • Principle through an example • Set-Theoretic concepts and Notations • Summing consecutive integers • The Product Principle • The principle • An application: Two-Elements subsets

25. Matrix Multiplication

29. Product Principle in General • Si and Sj are disjoint, |Si| = n • S = S1 U S2 U … U Sm • |S| = m |Si| = mn

30. Outline • Introduction to Counting • The Sum Principle • Principle through an example • Set-Theoretic concepts and Notations • Summing consecutive integers • The Product Principle • The principle • An application: Two-Elements subsets

33. First: Ordered Pairs

34. Now: Two-Element Subsets

35. Outline • Introduction to Counting • The Sum Principle • Principle through an example • Set-Theoretic concepts and Notations • Summing consecutive integers • The Product Principle • The principle • An application: Two-Elements subsets