1. Counting Techniques

2. Multiplication Principle (also called the Fundamental Counting Principle) Combinations Permutations Number of subsets of a given set

3. Suppose there are n different decisions and each decision has choices where i is some number from 1 to n. Then the overall total number of ways in which those n decisions can be made is the product Multiplication Principle

4. Examples Suppose you have 7 different shirts, 5 different pairs of pants and 3 pairs of shoes. How many outfits are possible? Suppose there are 10 questions on a multiple-choice exam and each question can be answered in 5 different ways (A, B, C, D or E). How many ways are there to complete the exam assuming every question is answered?

5. Examples Suppose you have 7 different shirts, 5 different pairs of pants and 3 pairs of shoes. How many outfits are possible? –Answer: 7*5*3=105 possible outfits Suppose there are 10 questions on a multiple-choice exam and each question can be answered in 5 different ways (A, B, C, D or E). How many ways are there to complete the exam assuming every question is answered? –Answer: ways to complete the exam.

6. Combinations The number of ways of choosing r distinct objects from n distinct objects is given by the formula Note and 0! = 1

7. Examples How many ways can 3 movies be chosen from a list of 5 movies? A committee consists of 10 people. How many ways are there to form a coalition of 5 people from the committee?

8. Examples How many ways can 3 movies be chosen from a list of 5 movies? –Answer: A committee consists of 10 people. How many ways are there to form a coalition of 5 people from the committee? –Answer:

9. Permutations The number of ways of selecting r distinct objects from n distinct objects and rearranging those r objects is given by the formula

10. Examples Suppose there are 10 movies playing in the theater. How many ways are there of selecting and ranking your favorite 3? There are 5 people in a coalition of voters. How many ways are there to rearrange those 5 people in distinct orderings?

11. Examples Suppose there are 10 movies playing in the theater. How many ways are there of selecting and ranking your favorite 3? –Answer: There are 5 people in a coalition of voters. How many ways are there to rearrange those 5 people in distinct orderings? –Answer: 5! = 120 ways

12. Given a set with n elements, the number of subsets of the given set is. Examples: –Let A = {x, y, z}. How many subsets does A have? –Suppose a committee consists of 3 people. How many possible coalitions can be formed from this committee? Number of Subsets

13. Examples: –Let A = {x, y, z}. How many subsets does A have? Answer: subsets –Suppose a committee consists of 3 people. How many possible coalitions can be formed from this committee? Answer: coalitions Number of Subsets