Learning Objectives for Section 7.3 Basic Counting Principles

1. Learning Objectives for Section 7.3 Basic Counting Principles After this lesson, you should be able to • apply and use the addition principle. • draw and interpret Venn diagrams. • apply and use the multiplication principle.

2. 7.3 Basic Counting Principles In this section, we will see how set operations play an important role in counting techniques.

3. Both Psych and English Solution to Opening Example In a certain class, there are 23 majors in Psychology, 16 majors in English and 7 students who are majoring in both Psychology and English. a) If there are 50 students in the class, how many students are majoring in neither of these subjects? b) How many students are majoring in Psychology alone?

4. Addition Principle (for Counting) This statement says that the number of elements in the union of two sets A and B is the number of elements of A plus the number of elements of B minus the number of elements that are in both A and B (because we counted those twice).

5. Both Psych and English The Addition Principle Applied 23 psychology majors; 16 English majors; 7 both psychology & English majors To find the number of students who are majoring in psychology or English: 23 + 16 – 7 = 32

6. A Second Problem Example: A survey of 100 college faculty who exercise regularly found that 45 jog, 30 swim, 20 cycle, 6 jog and swim, 1 jogs and cycles, 5 swim and cycle, and 1 does all three. • How many of the faculty members do not do any of these three activities? • How many just jog?

7. Solution J = Joggers S = Swimmers C = Cyclists We will start with the intersection of all three circles. This region represents the number of faculty who do all three activities. Then, we will proceed to determine the number of elements in each intersection of exactly two sets. 1 does all 3 J S C

8. Solution J = JoggersS = SwimmersC = Cyclists • A survey of 100 college faculty who exercise regularly found that: • 45 jog • 30 swim • 20 cycle • 6 jog and swim • 1 jogs and cycles • 5 swim and cycle • 1 does all three 1 does all 3 J S C

9. Example Example: Use a Venn diagram to answer the following: If n(A) = 12, n(B) = 27, and n(A B) = 30. What is n(AB)?

10. The Four Disjoint Sets of a Venn Diagram A B 2 1 4 3

11. A B Example Example: Label the four disjoint sets in the Venn diagram given: n(A) = 35, n(B) = 75, n(A  B ) = 95, and n(U) = 120

12. Example: Fred has 4 pairs of pants (black, tan, gray, and navy), 3 different shirts (plaid, stripe, and woven), and 2 pairs of shoes (dress and casual). How many different pants/shirt/shoe combinations can Fred make? Multiplication Principle Example*

13. continued Pants: black, tan, gray, and navy Shirts: plaid, stripe, and woven Shoes: dress and casual Tree Diagram:

14. Generalized Multiplication Principle Suppose that a task can be performed using two or more consecutive operations. If the first operation can be accomplished in m ways and the second operation can be done in n ways, the third operation in p ways and so on, then the complete task can be performed in m·n·p … ways.

15. More Examples Example: How many different ways can a team consisting of 28 players select a captain and an assistant captain? Example: A film critic is asked to rank 8 movies from first to last. How many rankings are possible? Example: A person rolls a six-sided die, and then flips a coin. What are the possible outcomes?

16. More Examples Example: How many different 5-letter code words are possible from the first 7 letters of the alphabet if • no letter is repeated? b) Adjacent letters must be different?

17. More Examples Example: How many different 10-digit telephone numbers are possible if the first digit cannot be a 0, 1, or 9?