Distributions

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Distributions
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Basic Model for Distributions of Distinct Objects

• The following problems are equivalent
• Distributing n distinct objects into b distinct
  boxes
• Stamping 1 of the b different box numbers on each
  of the n distinct objects.
• There are bn such distributions.
• If bi objects go in box i,
• then there are P(n b1, b2, , bb) distributions.

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Basic Model for Distributions of Identical Objects

• The following problems are equivalent
• Distribute n identical objects into b distinct
  boxes
• Draw n objects with repetition from b object
  types.
• There are (n b - 1)Cn such distributions of the
  n identical objects.

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Example 1

• A quarterback of a football team has a repertoire
  of 20 plays, and executes 60 plays per game.
• A frequency distribution is a graph of how many
  time each play was called during a game.
• How many frequency distributions are there?

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Example 2

• How many ways are there to assign 1,000 Justice
  Department lawyers to 5 different antitrust
  cases?
• How many, if 200 lawyers are assigned to each
  case?

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Example 3

• How many ways are there to distribute 40
  identical jelly beans among 4 children
• Without restriction?
• With each child getting 10 beans?
• With each child getting at least 1 bean?

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Example 3

• How many ways are there to distribute 40
  identical jelly beans among 4 children
• Without restriction?
• (40 4 - 1)C40
• With each child getting 10 beans?
• 1
• With each child getting at least 1 bean?
• (40 - 4 4 - 1)C(4 - 1)

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Example 4

• How many ways are there to distribute
• 18 chocolate doughnuts
• 12 cinnamon doughnuts
• 14 powdered sugar doughnuts
• among 4 policeman, if each policeman gets at
  least 2 doughnuts of each kind?

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Example 4

• It is the same number of ways to distribute
• 18 - 8 chocolate doughnuts
• 12 - 8 cinnamon doughnuts
• 14 - 8 powdered sugar doughnuts
• among 4 policeman without restriction.

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Example 4

• It is the same number of ways to distribute among
  4 policeman without restriction
• 18 - 8 chocolate doughnuts C(10 4 - 1, 4 - 1)
• 12 - 8 cinnamon doughnuts C(4 4 - 1, 4 - 1)
• 14 - 8 powdered sugar doughnuts C(6 4 - 1, 4 -
  1)

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Example 5

• How many ways are there to arrange the 26 letters
  of the alphabet so that no pair of vowels appear
  consecutively? (Y is
  considered a consonant).

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Example 5

• How many ways are there to arrange the 26 letters
  of the alphabet with no pair of vowels appearing
  consecutively? (Y is a consonant).
• There are 6 boxes around the vowels.
• The interior 4 have at least 1 consonant.
• Use the product rule
• Arrange the vowels 5!
• Distribute the consonant positions among the 6
  boxes
• C(21 - 4 6 - 1, 6 - 1)
• Arrange the consonants 21!

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Example 6

• How many integer solutions are there to
• x1 x2 x3 0, with xi ? -5?

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Example 6
How many integer solutions are there to x1 x2
x3 0, with xi ? -5? The same as that for x1
x2 x3 15, with xi ? 0.
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Example 7

• How many ways are there to distribute k balls
  into n distinct boxes (k lt n) with at most 1 ball
  in any box, if
• The balls are identical?
• The balls are distinct?

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Example 8

• How many arrangements of MISSISSIPPI are there
  with no consecutive Ss?

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Example 8

• How many arrangements of MISSISSIPPI are there
  with no consecutive Ss?
• There are 5 boxes around the 4 Ss.
• The middle 3 have at least 1 letter.
• Use the product rule
• Distribute the positions of the non-S letters
  among the 5 boxes.
• Arrange the non-S letters.

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Example 9

• How many ways are there to distribute 8 balls
  into 6 distinct boxes with the 1st 2 boxes
  collectively having at most 4 balls, if
• The balls are identical?

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Example 9

• How many ways are there to distribute 8 balls
  into 6 distinct boxes with the 1st 2 boxes
  collectively having at most 4 balls, if
• The balls are identical?
• Partition the distributions into sets where the
  1st 2 boxes have exactly k balls, for k 0, , 4.

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Example 9

• How many ways are there to distribute 8 balls
  into 6 distinct boxes with the 1st 2 boxes
  collectively having at most 4 balls, if
• The balls are distinct?

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Example 9

• How many ways are there to distribute 8 balls
  into 6 distinct boxes with the 1st 2 boxes
  collectively having at most 4 balls, if
• The balls are distinct?
• Partition the distributions into sets where the
  1st 2 boxes have exactly k balls, for k 0, ,
  4.
• For each k
• pick the balls that go into the 1st 2 boxes
• distribute them
• distribute the 8 - k other balls into the other 4
  boxes.