Lecture 10 Permutations and Combinations June 23, 200

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Lecture 10 Permutations and Combinations

• June 23, 2003

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agenda

• Permutations
• Combinations
• Some derivation of Permutations and Combinations
• Eliminating Duplicates
• r-Combinations with Repetitions

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Permutations

• Sample questions
• Five athletes (Amazon, Bobby, Corn, Dick and
  Ebay) compete in an Olympic event. Gold, silver
  and bronze medals are awarded in how many ways
  can the awards be made?

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Permutation (cont.)

• Order matters !!!
• The case that Amazon wins gold and Ebay wins
  silver is different from the case Ebay wins gold
  and Amazon wins silver.
• If the order is of significance, the
  multiplication rules are often used when several
  choices are made from one and the same set of
  objects.

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Permutations--Definition

• In general, if r objects are selected from a set
  of n objects, any particular arrangement of these
  r objects(say, in a list) is called a
  permutation.
• In other words, a permutation is an ordered
  arrangement of objects.
• By multiple principle, the total number of
  permutations of r objects selected from a set of
  n objects is n(n-1)(n-2)(n-r1)

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Permutations More examples

• Examples
• How many permutations of 3 of the first 5
  positive integers are there?
• How may permutations of the characters in
  COMPUTER are there? How many of these end in a
  vowel?
• How many batting orders are possible for a
  nine-man baseball team?

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Permutations - Calculation

• Background-Factorial notation
• 1!1, 2!(2)(1)2, 3!(3)(2)(1)6
• In general, n! n(n-1)(n-2) 321 for any
  positive integer n.
• It is customary to let 0!1 by definition.
• Calculation of Permutation

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Permutations -- Special Cases

• P(n,0)
• Theres only one ordered arrangement of zero
  objects, the empty set.
• P(n,1)
• There are n ordered arrangements of one object.
• P(n,n)
• There are n! ordered arrangements of n distinct
  objects (multiplication principle)

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Combinations

• An NBA team has 12 players, in how ways we can
  choose 5 from 12?
• Can we use permutations?
• Are we interested in the order of the players?

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Combinations (cont.)

• A combination is the same as a subset.
• When we ask for the number of combinations of r
  objects chosen from a set of n objects, we are
  simply asking How many different subsets of r
  objects can be chosen from a set of n objects?
• The order does not matter.

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Combinations (cont.)

• Any r objects can be arranged among themselves in
  r! permutations, which only count as one
  combination.
• So the n(n-1)(n-2)??(n-r1) different
  permutations of r objects chosen from a set of n
  objects contain each combination r! times.

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Combinations -- Definition

• The number of combinations of r objects
• chosen from a set of n objects is
• for r0,1,2,,n
• Or
• Other notations for C(n,r) are

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Combinations (cont.)

• For each combination, there are r! ways to
  permute the r chosen objects.
• Using the multiplication principle
• C(n,r)r!P(n,r)

are refer as binomial coefficients
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Combinations More examples

• In how many ways a committee of five can be
  selected from among the 80 employees of a
  company?
• In how many ways a research worker can choose
  eight of the 12 largest cities in the United
  States to be included in a survey?

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Combinations (cont.)

• Lets introduce a simplification
• When we choose r objects from a set of n
  objects we leave (n-r) of the n objects, so there
  are as many ways of leaving (or choosing) (n-r)
  objects as there are of choosing r objects.
• So for the solution of the previous problem, we
  have

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Combinations -- Special Cases
there is only one way to chose 0 objects from the
n objects

• C(n,0)
• C(n,1)
• C(n,n)

there are n ways to select 1 object from n objects
there is only one way to select n objects from n
objects, and that is to choose all the objects
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Permutations or Combinations ?

• There are fewer ways in a combinations problem
  than a permutations problem.
• The distinction between permutations and
  combinations lies in whether the objects are to
  be merely selected or both selected and ordered.
  If ordering is important, the problem involves
  permutations if ordering is not important the
  problem involves combinations.
• C(n,r) can be used in conjunction with the
  multiplication principle or the addition
  principle.
• Thinking of a sequence of subtasks may seem to
  imply ordering bit it just sets up the levels of
  the decision tree, the basis of the
  multiplication principle.
• Check the Fig 3. 9 to get an idea about the
  difference between permutation and combination.

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Eliminating duplicate

• A committee of 8 students is to be selected from
  a class consisting of 19 freshmen and 34
  sophomores. In how many ways can a committee with
  at least 1 freshman be selected?
• How many distinct permutations are there of the
  characters in the word Mongooses?
• How many distinct permutations are there of the
  characters in the word APALACHICOLA?

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Eliminating duplicate (cont.)

• In general, suppose there are n objects of
  which a set of n1 are indistinguishable for each
  other, another set of n2 are indistinguishable
  from each other, and so on, down to nk objects
  that are indistinguishable from each other. The
  number of distinct permutations of the n objects
  is

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r- Combinations with Repetitions

• A jeweler designing a pin has decided to use two
  stones chosen from diamonds, rubies and emeralds.
  In how many ways can the stones be selected?
• Answer-- D,R, D,D, D,E, E,R,E,E, R,R.
• Any other way to solve this problem? What if he
  needs five stones?

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r-Combinations with Repetitions(cont.)

• Some hints?
• 1 diamond, 3 rubies and 1 emerald
• 5 diamond, 0 rubies and 0 emerald
• 0 diamond, 5 rubies and 0 emerald
• 0 diamond, 0 rubies and 5 emerald
• What is it? Choose 5 stars from 7 elements, i.e.,
  C(7,5)

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r-Combinations with Repetitions (cont.)

• In general, there must be n-1 markers to indicate
  the number of copies of each of the n objects.
• We will have r (n-1) slots to fill (objects
  markers).
• We want the number of ways to select r out of the
  previous slots to fill.
• Therefore we want
• Six children use one lollipop each from a
  selection of red, yellow, and green lollipops. In
  how many ways can this be done?

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Summary
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Assignment

• Exercise 3.4--- 6, 10, 14, 24-28, 64, 72