Distributions
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.
Basic Model for Distributions of
Identical Objects
The following problems are equivalent:
• Distributing n identical objects into b distinct
boxes
• Picking a subset of n box numbers with
repetition from the b boxes numbers.
• There are (n + b - 1)Cn such distributions of
the n identical objects.
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?
Example 2
• How many ways are there to assign 1,000
“Justice” Department lawyers to 5 different
antitrust cases?
• How many, if 1,000/5 lawyers are assigned
to each case?
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)
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
demands at least 2 doughnuts of each kind?
C(18 - 8 + 4 - 1, 4 - 1) C(12 - 8 + 4 - 1, 4 - 1)
C(14 - 8 + 4 - 1, 4 - 1)
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).
• There are 6 boxes around the vowels.
• The interior 4 have at least 1 consonant.
• Using the product rule:
• Arrange the vowels: 5!
• Distribute the consonant positions into the 6 boxes:
C(21 - 4 + 4 - 1, 4 - 1)
• Arrange the consonants: 21!
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.
Example 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?
Example 8
• How many arrangements of MISSISSIPPI
are there with no pair of 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.
Example 9
• How many ways are there to distribute 8
balls into 6 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.
• 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.
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