Chapter 8 Counting Principles: Further Probability Topics

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Chapter 8Counting Principles Further
Probability Topics

• Section 8.1
• The Multiplication Principle
• Permutations

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Warm Up for Sections 8.1 and 8.2

• A certain game at an amusement card consists of
  a person spinning a spinner, choosing a card, and
  then tossing an unbiased coin. Prizes are
  awarded based on the combination created from
  performing each of the three tasks.
• The spinner has three equal areas represented by
  Purple, Gold, and Red the cards to choose from
  include a King, Queen, and Joker and the coin
  has a Crown on one side and a Donkey on the
  other.
• How many possible outcomes are there?
• If the order in which the tasks were performed
  made a difference, would there be more outcomes
  or fewer outcomes?

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• Alice cant decide what to wear between a pair of
  shorts, a pair of pants, and a skirt. She has
  four tops that will go with all three pieces
  one red, one black, one white, and one striped.
• How many different outfits could Alice create
  from these items of clothing?

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Bottom Top Outfit
Shorts, Red Top
Red Black White Striped Red Black White Striped
Red Black White Striped
Shorts, Black Top
Shorts, White Top
Shorts
Shorts, Striped Top
Pants
12 outfits!!
Skirt
If the tree diagram is finished, how many outfits
will she have?
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• Tree diagrams are not often convenient, or
  practical, to use when determining the number of
  outcomes that are possible.
• Rather than using a tree diagram to find the
  number of outfits that Alice had to choose from,
  we could have used a general principle of
  counting the multiplication principle.

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Multiplication Principle
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• Alice cant decide what to wear between a pair of
  shorts, a pair of pants, and a skirt. She has
  four tops that will go with all three pieces
  one red, one black, one white, and one striped.
• How many different outfits could Alice create
  from these items of clothing?
• Using the multiplication principle, we multiply
  the number of options she has for what to wear on
  bottom and the number of options she has for
  what to wear on top.
• 3 bottoms 4 tops 12 outfits

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• A product can be shipped by four airlines and
  each airline can ship via three different routes.
  How many distinct ways exist to ship the product?

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• How many different license plates can be made if
  each license plate is to consist of three letters
  followed by three digits and replacement is
  allowed?
• ___ ___ ___ ___ ___ ___
• L L L D D D
• If replacement is not allowed?

26
26
26
10
10
10
26 ³ 10 ³
17, 576, 000
26 25 24 10 9
8
___ ___ ___ ___ ___ ___ L L
L D D D
11, 232, 000
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How many different license plates can be made if
each license plate begins with 63 followed by
three letters and two digits?
How many different social security numbers are
possible if the first digit may not be zero?
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Marie is planning her schedule for next semester.
She must take the following five courses
English, history, geology, psychology, and
mathematics. a.) In how many different ways can
Marie arrange her schedule of
courses? b.) How many of these schedules have
mathematics listed first?
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You are given the set of digits 1, 3, 4, 5,
6. a.) How many three-digit numbers can be
formed? b.) How many three-digits numbers can
be formed if the number must be
even? c.) How many three-digits numbers can be
formed if the number must be even and no
repetition of digits is allowed?
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• A certain Math 110 teacher has individual photos
  of each of her three dogs Indy, Sam, and Jake.
  In how many ways can she arrange these photos in
  a row on her desk?

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Factorial Notation
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If seven people board an airplane and there are
nine aisle seats, in how many ways can the people
be seated if they all choose aisle seats?
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Permutations

• A permutation of r (where r 1) elements from
  a set of n elements is any specific ordering or
  arrangement, without repetition, of the r
  elements.
• Each rearrangement of the r elements is a
  different permutation.
• Permutations are denoted by nPr or P(n, r)
• Clue words arrangement, schedule, order,
  awards, officers

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• A disc jockey can play eight records in a
  30-minute segment of her show. For a particular
  30-minute segment, she has 12 records to select
  from. In how many ways can she arrange her
  program for the particular segment?
• A chairperson and vice-chairperson are to be
  selected from a group of nine eligible people.
  In how many ways can this be done?

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Distinguishable Permutations

• If the n objects in a permutation are not all
  distinguishable that is, if there so many of
  type 1, so many of type 2, and so on for r
  different types, then the number of
  distinguishable permutations is
• n! .
• n ! n ! n !

r
1
2
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How many distinct arrangements can be formed from
all the letters of SHELTONSTATE?
Step 1 Count the number of letters in the word,
including repeats.
12 letters
Step 2 Count the number of repetitious letters
and the number of times each
letter repeats.
S 2 repeats E 2 repeats T
3 repeats
Solution 12! .
2! 2! 3!
19, 958, 000
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• In how many distinct ways can the letters of
  MATHEMATICS be arranged?
• In how many distinct ways can the letters of
  BUCCANEERS be arranged?