Topic III: Counting Rules

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Topic III Counting Rules
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Counting Rules

• There are 5 main counting rules
• The Multiplication Principle
• The Permutation Principle
• The Permutation of n objects taken r at a
  time Principle
• The Partition Principle
• The Combination Principle

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The Multiplication Principle

• If you choose one item from a group of M items
  and another item from a group of N items, the
  total number of two-item choices is M N
• If there are r different groups each with nr
  items, then the total number of r-item choices is
  n1 n2 . . . nr

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Example 3

• A particular brand of women's jeans can be
  ordered in seven different sizes, three different
  colours and three different styles. How many
  jeans would have to be ordered if a store wanted
  to have one pair of each type?

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Example 4

• At a restaurant you can choose among 5
  appetizers, 34 main dishes, and 10 desserts.
• What is the total number of different meals you
  can choose from each, including one appetizer,
  one main dish, and one dessert.

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Special Case of the Multiplication Principle

• If any one of k different mutually exclusive and
  collectively exhaustive events can occur on each
  of n trials, then the number of possible outcomes
  is equal to

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Example 5

• If a coin is tossed seven times, how many
  different outcomes are possible?
• If a die is tossed seven times, how many
  different outcomes are possible.

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The Permutation Principle I

• A permutation is a special case of the
  multiplication principle
• A Permutation of a finite sequence S is any
  finite sequence obtained by rearranging the terms
  of S in some order.
• A permutation is an arrangement
• Used when Order matters or there is an assignment
• Repetitions are not allowed (unless there is
  repetition in the input)
• The formula used to find permutations is n!, the
  total number of arrangements

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The Permutation Principle II

• If there are n members of a group of items then
  the first can be arranged (n) different ways, for
  each of these the second can be arranged (n-1)
  different ways and for each of the first two the
  third can be done (n-2) ways etc.
• For example, if a set of 4 textbooks is to be
  placed on a shelf, any of the four books could
  occupy the first position, after the first
  position is filled there are three books and any
  of then could occupy the second position and so
  on, this assignment procedure is continued until
  all the positions are occupied.

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Example 6

• There are five teams in Group C of the Manning
  Cup football Competition. How many different
  orders of finish are there for the five teams?

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Example 7

• A store manager has six display shelves available
  to place six different styles of womens blouses.
  Each blouse will be allowed one and only one
  shelf. How many ways are there to position these
  blouses on the shelves?

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The Permutation of n Objects Taken r at a
Time Principle I

• In some cases there may be n objects from which
  only r can be chosen
• Order is still important
• Each possible arrangement is called a permutation
• The number of permutations of n distinct objects
  taken r at a time is

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The Permutation of n Objects Taken r at a
Time Principle II

• For example, modifying the preceding
  illustration, suppose there four books involved,
  however the shelf can only hold 3 books. We may
  want to find out how many arrangements of three
  books are possible.

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Example 8

• Four names are drawn from among the 24 members of
  a club for the offices of president, vice
  president, treasurer and secretary. In how many
  different ways can this be done?

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The Partition Principle I

• Some cases may require partitioning n distinct
  objects into k distinct groups containing n1,
  n2, . . ., nk objects
• Each object must appear only in one group
• ?ni n
• The number of distinct arrangements N

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The Partition Principle II

• The number of distinct arrangements is
• n the number of distinct objects
• Rearrangements of objects within a group do not
  count

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Example 9

• There are 12 construction workers to be assigned
  to three different areas of Highway 2000. St.
  James is to be assigned 3 workers, Portmore 4
  workers and St. Elizabeth 5 workers. How many
  different ways could the assignment be made?

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Example 10

• A labour dispute has arisen concerning the
  distribution of 20 labourers to four different
  construction jobs. The first job (considered to
  be very undesirable) required 6 labourers the
  second, third fourth utilised 4, 5, and 5
  labourers respectively. The dispute arose over
  an alleged random distribution of the labourers
  to the jobs which placed all four members of a
  particular ethnic group on job 1. Determine the
  number of sample points in the sample space S for
  this experiment. That is, determine the number
  of ways the 20 labourers can be divided into
  groups of the appropriate sizes to fill all of
  the jobs.

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The Combination Principle I

• A combination simply refers to the number of
  subsets of r objects that can be selected from n
  distinct objects
• It is a special case of the Partition Principle
  where the number of partitions, k, is equal to 2
• Under the permutation principle, order was
  important, in situations where the order of the
  outcomes is not important and only the number of
  ways in which r objects can be selected out of n,
  then the combination principle is used.

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The Combination Principle II

• Order is not important (the same holds for
  partitions)
• The number of combinations of n distinct objects
  taken r at a time is
• Note this is sampling without replacement

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Example 11

• The daily lottery consists of selecting six
  winning numbers out of numbers. How many
  different combinations of winning numbers are
  possible?