MAT 2720 Discrete Mathematics

1. MAT 2720Discrete Mathematics Section 6.1 Basic Counting Principles http://myhome.spu.edu/lauw

2. General Goals • Develop counting techniques. • Set up a framework for solving counting problems. • The key is not (just) the correct answers. • The key is to explain to your audiences how to get to the correct answers (communications).

3. Goals • Basics of Counting • Multiplication Principle • Addition Principle • Inclusion-Exclusion Principle

4. Example 1 LLL-DDD License Plate # of possible plates = ?

5. Analysis LLL-DDD License Plate # of possible plates = ? Procedure: Step 1: Step 4: Step 2: Step 5: Step 3: Step 6:

6. Multiplication Principle Suppose a procedure can be constructed by a series of steps Number of possible ways to complete the procedure is

7. Example 2(a) Form a string of length 4 from the letters A, B, C , D, E without repetitions. How many possible strings?

8. Example 2(b) Form a string of length 4 from the letters A, B, C , D, E without repetitions. How many possible strings begin with B?

9. Example 3 Pick a person to joint a university committee. # of possible ways = ?

10. Analysis Pick a person to joint a university committee. # of possible ways = ? The 2 sets: :

11. Addition Principle • Number of possible element that can be selected fromX1or X2or …or Xkis • OR

12. Example 4 A 6-person committee composed of A, B, C , D, E, and F is to select a chairperson, secretary, and treasurer.

13. Example 4 (a) In how many ways can this be done?

14. Example 4 (b) In how many ways can this be done if either A or B must be chairperson?

15. Example 4 (c) In how many ways can this be done if E must hold one of the offices?

16. Example 4 (d) In how many ways can this be done if both A and D must hold office?

17. Recall: Intersection of Sets (1.1) The intersection of X and Y is defined as the set

18. Recall: Intersection of Sets (1.1) The intersection of X and Y is defined as the set

19. Example 5 What is the relationship between

20. Inclusion-Exclusion Principle

21. Example 4(e) How many selections are there in which either A or D or both are officers?.

22. Remarks on Presentations • Some explanations in words are required. In particular, when using the Multiplication Principle, use the “steps” to explain your calculations • A conceptual diagram may be helpful.

23. MAT 2720Discrete Mathematics Section 6.2 Permutations and Combinations Part I http://myhome.spu.edu/lauw

24. Goals • Permutations and Combinations • Definitions • Formulas • Binomial Coefficients

25. Example 1 6 persons are competing for 4 prizes. How many different outcomes are possible? Step 1: Step 2: Step 3: Step 4:

26. r-permutations A r-permutation of n distinct objects is an ordering of an r-element subset of

27. r-permutations A r-permutation of n distinct objects is an ordering of an r-element subset of The number of all possible ordering:

28. Example 1 6 persons are competing for 4 prizes. How many different outcomes are possible?

29. Theorem

30. Example 2 100 persons enter into a contest. How many possible ways to select the 1st, 2nd, and 3rd prize winner?

31. Example 3(a) How many 3-permutations of the letters A, B, C , D, E, and F are possible?

32. Example 3(b) How many permutations of the letters A, B, C , D, E, and F are possible. Note that, “permutations” means “6-permutations”.

33. Example 3(c) How many permutations of the letters A, B, C , D, E, and F contains the substring DEF?

34. Example 3(d) How many permutations of the letters A, B, C , D, E, and F contains the letters D, E, and F together in any order?