distributions. basic model for distributions of distinct objects the following problems are...
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![Page 1: Distributions. Basic Model for Distributions of Distinct Objects The following problems are equivalent: Distributing n distinct objects into b distinct](https://reader035.vdocument.in/reader035/viewer/2022062313/56649d6d5503460f94a4d057/html5/thumbnails/1.jpg)
DistributionsDistributions
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Basic Model for Distributions of Basic Model for Distributions of DistinctDistinct Objects 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 Basic Model for Distributions of IdenticalIdentical Objects 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 1Example 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 2Example 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 3Example 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 3Example 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 4Example 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 4Example 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 4Example 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 5Example 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 5Example 5How 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 6Example 6
How many integer solutions are there to
x1 + x2 + x3 = 0, with xi -5?
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Example 6Example 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 7Example 7
How many ways are there to distribute k balls
into n distinct boxes (k < n) with at most 1
ball in any box, if:
• The balls are identical?
• The balls are distinct?
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Example 8Example 8
How many arrangements of MISSISSIPPI are there with no consecutive Ss?
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Example 8Example 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 9Example 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 9Example 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 9Example 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 9Example 9How 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.