1. Distributions

2. 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.

3. 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.

4. 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?

5. 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?

6. 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?

7. Example 3
How many ways are there to distribute 40 identical jelly beans among 4 children:
Without restriction?
( )C40
With each child getting 10 beans? 1
With each child getting at least 1 bean?
( )C(4 - 1)

8. 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?

9. Example 4 It is the same number of ways to distribute:
chocolate doughnuts
cinnamon doughnuts
powdered sugar doughnuts
among 4 policeman without restriction.

10. Example 4
It is the same number of ways to distribute among 4 policeman without restriction :
chocolate doughnuts C( , 4 - 1)
cinnamon doughnuts C( , 4 - 1)
powdered sugar doughnuts C( , 4 - 1)

11. 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).

12. 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( , 6 - 1)
Arrange the consonants: 21!

13. Example 6 How many integer solutions are there to
x1 + x2 + x3 = 0, with xi  -5?

14. 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.

15. 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?

16. Example 8
How many arrangements of MISSISSIPPI are there with no consecutive Ss?

17. 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.

18. 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?

19. 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.

20. 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?

21. 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.