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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.

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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?

Both Psych and English

English majorsPsychology majors

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Addition Principle (for Counting)

( ) ( ) ( ) ( )n A B n A n B n A B 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).

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The Addition Principle Applied

Both Psych and English

English majorsPsychology majors

( ) ( ) ( ) ( )

( )

n P E n P n E n P E

n P E

23 psychology majors; 16 English majors; 7 both psychology & English majors

Find the number of students who are majoring in psychology or English:

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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.

a)How many of the faculty members do not do any of these three activities?

b)How many just jog?

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Solution

1 does all 3

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.

J

J = Joggers

S = Swimmers

C = Cyclists

C

S

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Solution

1 does all 3

J

J = JoggersS = SwimmersC = Cyclists

C

S

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

Fill in the remaining areas.

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Example

Example: Use the addition principle to answer the following. Then show the result using a Venn diagram.

If n(A) = 12, n(B) = 27, and n(A B) = 30. What is n(AB)?

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The Four Disjoint Sets of a Venn Diagram

1 2

3

4

A B

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Example

Example: Label the number of elements in the four disjoint sets in the Venn diagram given:

n(A) = 35, n(B) = 75, n(A B ) = 95, and n(U) = 120

A B

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Multiplication Principle Example*

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?

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continued

Pants: black, tan, gray, and navy

Shirts: plaid, stripe, and woven

Shoes: dress and casual

Tree Diagram:

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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.

Multiplication Principle

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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?

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PLEASE NOTE

There are 26 letters in the English alphabet: A, B, C, D, E, F, G, H, I, J, K, L, M, N, O ,P ,Q, R, S, T, U V, W, X, Y,

Z

There are 10 digits in the decimal system: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

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More Examples

Example: How many different 5-letter code words are possible using the first 7 letters of the alphabet if

a) letters can be repeated?

b) no letter is repeated?

c) Adjacent letters must be different?

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More Examples

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

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Examples from Text

Page 372 #2 – 12 even, 18, 24, 36, 42