1. Methods of Counting Outcomes BUSA 2100, Section 4.1

2. Counting Rules l Counting rules provide a way to determine the number of possible outcomes for a situation without having to list or count them all. l The first counting rule is called the Multiplication Principle. It applies to outcomes for which order matters, i.e. order makes a difference.

3. Multiplication Principle l Multiplication Principle: The total number of outcomes for an ordered situation is the product of the number of outcomes for each part of the situation. l Example 1: How many different phone numbers are possible with the same area code? l (Suppose first digit cannot be a zero.)

4. Multiplication Principle, p. 2 l Does order make a difference?

5. Multiplication Principle, p. 3 l Example 2: How many different license plates are possible using three numbers followed by three letters? l (Suppose zeros are not allowed and repetitions are not allowed for letters.)

6. Multiplication Principle, p. 4 l The Multiplication Principle is applicable whenever: (1) Order matters, i.e. objects in different orders represent different outcomes; l (2) Repetitions may or may not be allowed, depending upon the content of the problem.

7. Permutations l Definition: A permutation is an ordered arrangement of distinct objects (repetitions are not allowed). l Example 1: How many ways can 5 people line up? l Lines are ordered arrangements and the same person can’t be chosen twice (no repetitions). So we use permutations.

8. Permutations, Page 2 l Permutation problems are done in the same way as Multiplication Principle problems. l Permutations are a special case of the Multiplication Principle. l In a permutation, the numbers occur in descending order.

9. Permutations, Page 3 l What is the symbol for the product of the integers from 5 down to 1? l Ex. 2: How many ways can 3 people be selected from 7 people if the 1st person chosen is President, the 2nd is Vice President, and the 3rd is Secretary?

10. Combinations l Definition: A combination is a selection of distinct objects for which order is not important (does not matter). l Example 1: How many different committees of 3 people can be chosen from 7 people? l Is order important?

11. Combinations, Page 2 l For convenience, refer to the 7 people as A,B,C,D,E,F,G. Note that ABC, ACB, BAC, BCA, CAB, and CBA all refer to the same 3 people. l They represent six permutations, but only one combination.

12. Combinations, Page 3 l Summary: If order matters, use the Multip. Principle or permutations; if order doesn’t matter, use combinations.