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 b n such distributions. If b i objects go in box i, then there are P(n; b 1, b 2, …, b b ) distributions.

3. 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) C n 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 1,000/5 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? (40 + 4 - 1) C 40 With each child getting 10 beans? 1 With each child getting at least 1 bean? (40 - 4 + 4 - 1) C (4 - 1)

7. 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)

8. 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!

9. Example 6 How many integer solutions are there to x 1 + x 2 + x 3 = 0, with x i  -5? The same as that for x 1 + x 2 + x 3 = 15, with x i  0.

10. 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?

11. 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.

12. 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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