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: 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 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 200 lawyers are assigned to each case?

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?

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 gets at least 2 doughnuts of each kind?

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.

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)

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).

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!

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

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 consecutive Ss?

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.

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?

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.

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?

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.