DistributionsDistributions

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.

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.

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?

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?

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?

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)

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?

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.

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)

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).

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!

Example 6Example 6

How many integer solutions are there to

x1 + x2 + x3 = 0, with xi -5?

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.

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?

Example 8Example 8

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

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.

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?

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.

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?

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.