Learning Objectives for Section 7'3 Basic Counting Principles

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Learning Objectives for Section 7.3 Basic
Counting Principles

• The student will be able to apply and use the
  addition principle.
• The student will be able to draw and interpret
  Venn diagrams.
• The student will be able to apply and use the
  multiplication principle.

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Addition Principle (For Counting)

• The number of elements in a set A is denoted by
  n(A).

• We introduce the following principle of counting
  that can be illustrated using a Venn diagram.
• 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).

• Sets A and B are called disjoint if A?B ?.
• If A and B are disjoint, then n(A?B) n(A)
  n(B).

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• Example 1 According to a survey of business
  firms in Cypress, 345 firms offer their employees
  group life insurance, 285 offer long-term
  disability insurance, and 115 offer group life
  insurance and long-term disability insurance.

How many firms offer their employees group life
insurance or long-term disability insurance?
SOLUTION If G set of firms that offer
employees group life insurance, D set of firms
that offer employees long-term disability
insurance, and G?D set of firms that offer
group life insurance and long-term disability
insurance G?D set of firms that offer group
life insurance or long-term disability
insurance n(G) _____ n(D) _____ n(G?D)
_____ n(G?D) _____ _____ - _____ _____
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• Example 2 A small town has 2 radio stations, an
  AM station and an FM station. A survey of 100
  residents of the town produced the following
  results In the last 30 days, 65 people have
  listened to the AM station, 45 people have
  listened to the FM station, and 30 have listened
  to both stations.

Let U the group of people surveyed Let A set
who listened to AM station Let F set who
listened to FM station n(A) _____ n(F)
_____ n(A?F) _____ n(A?F) _____ _____ -
_____ _____
(A) How many people in the survey have listened
to the AM station but not to the FM station?
_____ (B) How many have listened to the FM
station but not to the AM station? _____ (C) How
many have not listened to either station? _____
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Multiplication Principlefor Counting

• Multiplication Principle for Counting If two
  operations, O1 and O2 are performed in order,
  where O1 has N1 possible outcomes and O2 has N2
  possible outcomes, then there are N1N2 possible
  outcomes of the first operation followed by the
  second.
• In general, if n operations O1, O2, On are
  performed in order, with possible number of
  outcomes N1, N2, Nn respectively, then there
  are N1N2 Nn possible combined outcomes of the
  operations performed in the given order.

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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 mnp ways.

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Multiplication PrincipleAn Example

• 3. An Apple store stocks 4 types of ipod
  ipod4G, ipod8G, mini, and nano. They are low on
  stock and are only available in blue and red.
  What are the combined choices, and how many
  combined choices are there? Solve using a tree
  diagram.

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Multiplication PrincipleContinued

• If we had asked from the 26 letters of the
  alphabet, how many ways can 3 letters appear in a
  row on a license plate is no letter is
  repeated?, it would be tedious to list the
  possibilities in a tree diagram so we would use
  the multiplication counting principle to solve
  this problem. What would the answer be?
• 4. Each question on your multiple choice
  computer work has 4 choices. There are 20
  questions on this weeks work. How many
  different combinations of answers exist for the
  20 questions?

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• 5. How many 4-letter code words are possible
  using the first 10 letters of the alphabet if
• No letter can be repeated?
• Letters can be repeated?
• Adjacent letters cannot be alike?

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Multiplication PrincipleAn Example

• To illustrate this principle, lets start with an
  example.
• Suppose you have 4 pairs of trousers in your
  closet, 3 different shirts and 2 pairs of shoes.
  Assuming that you must wear trousers (we hope
  so!), a shirt and shoes, how many different ways
  can you get dressed?

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Multiplication PrincipleExample Solution

• Lets assume the colors of your pants are black,
  grey, rust, olive. You have four choices here.
  The shirt colors are green, marine blue and dark
  blue. For each pair of pants chosen (4) you have
  (3) options for shirts. You have 12 43 options
  for wearing a pair of trousers and a shirt.
• Now, each of these twelve options, you have two
  pair of shoes to choose from (Black or brown).
  Thus, you have a total of 432 24 options to
  get dressed.

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Another Problem

• How many different ways can a team consisting of
  28 players select a captain and an assistant
  captain?

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Another Problem

• How many different ways can a team consisting of
  28 players select a captain and an assistant
  captain?
• Solution
• Operation 1 Select the captain. If all team
  members are eligible to be a captain, there are
  28 ways this can be done.
• Operation 2 Select the assistant captain.
  Assuming that a player cannot be both a captain
  and assistant captain, there are 27 ways this can
  be done, since there are 27 team members left who
  are eligible to be the assistant captain.
• Using the multiplication principle, there are
  (28)(27) 756 ways to select both a captain and
  an assistant captain.

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Extra Example

• To see how sets play a role in counting, consider
  the following 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.
• If there are 50 students in the class, how many
  students are majoring in neither of these
  subjects?
• How many students are majoring in Psychology
  alone?

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Solution(continued)
7 students in this region
Both Psych and English
23 16 7 32
Do you see how the numbers of each region are
obtained from the given information in the
problem? We start with the region represented by
the intersection of Psych and English majors (7).
Then, because there are 23 Psych majors, there
must be 16 Psych majors remaining in the rest of
the set. A similar argument will convince you
that there are 9 students who are majoring in
English alone.
9 students in this region
16 students here
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A Second Problem

• 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?
• We will solve this problem using a three-circle
  Venn Diagram in the accompanying slides.

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Solution
We will start with the intersection of all three
circles. This region represents the number of
faculty who do all three activities (one). Then,
we will proceed to determine the number of
elements in each intersection of exactly two sets.
1 does all 3
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Solution(continued)

• Starting with the intersection of all three
  circles, we place a 1 in that region (1 does all
  three). Then we know that since 6 jog and swim so
  5 faculty remain in the region representing those
  who just jog and swim. Five swim and cycle, so 4
  faculty just swim and cycle but do not do all
  three. Since 1 faculty is in the intersection
  region of joggers and cyclists, and we already
  have one that does all three activities, there
  must be no faculty who just jog and cycle.

Since the sum of the numbers of these disjoint
regions is 84, there must be 16 faculty who do
none of these activities.
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Final Example

• A sportswriter is asked to rank 8 teams in the
  NBA from first to last. How many rankings are
  possible?

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Final Example

• A sportswriter is asked to rank 8 teams in the
  NBA from first to last. How many rankings are
  possible?
• Solution There are 8 choices that can be made
  for the first place team since all teams are
  eligible. That leaves 7 choices for the second
  place team. The third place team is determined
  from the 6 remaining choices and so on.
• Total is the product of 8(7)(6)(5)(4)(3)(2)(1)
  40,320.