Introduction to Counting Methods MATH 102 Contemporary Math S. Rook

Overview Section 13.1 in the textbook: – Systematic counting – Revisiting tree diagrams

Systematic Counting

Suppose we want to count a set of objects For simple sets, list the elements of the set – e.g. Pick one person out of 10 for a job, pick one card from a standard deck of cards, flip one coin For more complicated sets, the best way to count is systematically – i.e. In an organized manner Think of more complicated counting problems as occurring in stages – i.e. One step at a time

Systematic Counting (Example) Ex 1: Assume you are rolling two dice, the first is red and the second is green. Use a systematic listing to determine the number of ways to: a) Roll a total of 5 b) Roll a total less than 6 c) Roll a total of more than 9

Revisiting Tree Diagrams

Recall tree diagrams from Section 1.1 – An effective way to list possibilities for events that occur in stages Consider listing two pair elements of set S = {A, B, C} Two other conditions can affect counting problems: – Does order matter? If yes, AB and BA are two different elements If no, AB and BA are the same element – Is repetition allowed? If yes, AA is an element If no, AA is not an element

Revisiting Tree Diagrams (Example) Ex 2: A “best of 3” series in sports ends when a team either wins 2 of 3 games or loses 2 of 3 games. Suppose that the only results of each game are winning and losing. a) Draw a tree diagram representing all outcomes of a “best of 3” series from the vantage point of a single team. b) How many total outcomes exist?

Summary After studying these slides, you should know how to do the following: – Count in a systematic fashion – Apply tree diagrams to counting problems Additional Practice: – See problems in Section 13.1 Next Lesson: – The Fundamental Counting Principle (Section 13.2)