Transcript 6.3 Basic Counting Principles

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
Psychology majors
English 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
23 psychology majors; 16 English majors; 7 both psychology & English majors
Find the number of students
who are majoring in psychology
or English:
Both
Psych
and
English
n ( P  E )  n ( P )  n( E )  n ( P  E )
Psychology majors
English majors
n( P  E ) 


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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
J = Joggers
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.
S = Swimmers
C = Cyclists
1 does all 3
J
S
C
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Solution
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.
J = Joggers
S = Swimmers
C = Cyclists
1 does all 3
J
S
C
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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(AB)?
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The Four Disjoint Sets of a Venn
Diagram
A
B
2
1
4
3
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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?
Pants: black, tan, gray, and navy
Shirts: plaid, stripe, and woven
Shoes: dress and casual
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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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How many different 5-letter code words are possible using the
first 7 letters of the alphabet if adjacent letters must be different?
ABCDEFG
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Examples from Text
 Page 372 #2 – 12 even, 18, 24, 36, 42
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