Lecture 9

1. Lecture 9 June 18, 2003

2. Combinatorics The branch of mathematics studying the enumeration, combination, and permutation of sets of elements and the mathematical relations which characterize these properties. Examples of combinatorial problems: Scheduling classes (or tournament play); Determining cheapest route for a multi-city tour; Tracing a figure without picking up your pencil or repeating a line; Determining the odds of a full house in poker; Designing reliable networks; Creating efficient computer programs.

3. Examples Two dice are rolled, one black and one white, how many different rolls are possible? (white 1 is not black 1) How many ways are there to choose three officers from a club of 25 people?

4. Multiplication Principle If a choice consists of two steps, of which one can be made in m ways and the other in n ways, then the whole choice can be made in m*n ways. Example--- For any finite set S, let |S| denote the number of elements in S. If A and B are finite sets, then |AXB|=|A|*|B|

5. Multiplication Principle So we can generalize the multiplication principle for the �multiplication of choices� so that it applies to choices involving more than two steps. If in a sequence of n decisions, the number of choices for decision i does not depend on how the previous i-1 decisions are made, then the total number of ways to make the whole sequence of decisions is the product of the number of choices for each decision. So for k steps, where k is a positive integer, we have: If a choice consists of k steps, of which the first can be made in n1 ways, the second in n2 ways, �, and the kth in nk ways, then the whole choice can be made in n1�n2���nk ways.

6. Addition Principle If a set can be partitioned into disjoint subsets, then the number of objects in the set =� the sum of the number of objects in each of its parts. Example : The number of people in this room equals the number of men plus the number of women.

7. Addition Principle(cont.) The number of subsets of {1,2,3,4} equals the number of 0-element subsets� =1 the number of 1-element subsets {1},{2}, {3},{4} =4 the number of 2-element subsets {1,2}, {1,3}, {1,4}, {2,3}, {2,4}, {3,4} =6 the number of 3-element subsets {1,2,3}, {1,2,4}, {1,3,4}, {2,3,4} =4 the number of 4-element subsets {1,2,3,4} =1 + TOTAL =16

8. Addition Principle(cont.) A child is allowed to choose one jellybean out of two jellybeans, one red and one black, and a gummy bear out of three gummy bears, yellow, green, and white. How many different sets of candy can the child have?

9. Using the principles together Now we can use both principles at the same time: If a woman has seven blouses, five skirts and nine dresses, how many different outfits does she have? (7*5)+9 How many three-digit integers are even?

10. Decision Trees If the multiplication principle doesn�t apply we can use less regular decision trees. We can use the multiplication principle, when the number of outcomes at any level of the tree is the same throughout that level. Number of outcomes of an event based on a series of possible choices. Example: Draw the decision tree for the number of strings X�s, Y�s and Z�s with length 3 that do not have a Z following a Y.

11. Principles in Set Now let�s see some properties: For any finite set S, let |S| denote the number of elements in S. If A and B are finite sets, then |AXB| = |A|�|B| Let A and B be disjoint finite sets. Then |AUB| = |A|+|B| If A and B are finite sets, then |A-B| = |A| - |A ? B| and |A-B| = |A| - |B| if B ? A

12. Principle of Inclusion and Exclusion If A and B are any subset of a universal set S, then {A-B}, {B-A}, {A ? B} are mutually disjoint sets. Venn diagram representation What is (A-B) ? (B-A) ? (A ? B)? And |(A-B) ? (B-A) ? (A ? B)| ? So finally |A ? B| = |A| + |B| - |A ? B|

13. Principle of Inclusion and Exclusion (cont.) So if we�re dealing with two sets: |A ? B| = |A| + |B| - |A ? B| And if we are dealing with three sets: |A ? B ? C| = |A| + |B| + |C| - |A ? B| - |A ? C| - |B ? C| + |A ? B ? C | What is the venn gram representation of it?

14. So the pattern is: If we have n sets, we should add the number of elements in the single sets, subtract the number of elements in the intersection of two sets, add the number of elements in the intersection of three sets, subtract the number of elements in the intersection of four sets, and so on. Principle of Inclusion and Exclusion Principle of Inclusion and Exclusion (cont.)

15. Principle of Inclusion and Exclusion (cont.) How to prove it? Mathmatical induction. Base case�� Assumption �.. Show �. (Page 205-206)

16. Principle of Inclusion and Exclusion (cont.) Example� A survey of 150 college students reveals that 83 own automobiles, 97 own bikes, 28 own motorcycles, 53 own a car and a bike, 14 own a car and a motorcycles, 7 own a bike and a motorcycle, and 2 own all three. Question� How many students own only a bike? How many students do not own any of the three?

17. Pigeonhole Principle If more than k items are placed into k bins, then at least one bin contains more than one item. Example� How many times must a single die be rolled in order to guarantee getting the same value twice?

18. Excercise Exercise 3.2� 28, 48, 49,58-64,69, 72 Exercise 3.3� 7, 11, 15, 19