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: • 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)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 1,000/5 lawyers are assigned to each case? 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 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) 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! 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 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. 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. Characters • • • • • •