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Distributions

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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. • Ifbi 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? (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)

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: • 18 - 8 chocolate doughnuts • 12 - 8 cinnamon doughnuts • 14 - 8 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 : • 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)

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(21 - 4 + 6 - 1, 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 exactlyk 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 exactlyk 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.