# 3: Distribution Problems

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• 3.1: The Idea of Distribution
It is helpful to have more than one way to think of solutions to problems. In the case of distribution problems, another popular model for distributions is to think of putting balls in boxes rather than distributing objects to recipients. Passing out identical objects is modeled by putting identical balls into boxes. Passing out distinct objects is modeled by putting distinct balls into boxes.
• 3.2: Partitions and Stirling Numbers
We have seen how the number of partitions of a set of k objects into n blocks corresponds to the distribution of k distinct objects to n identical recipients. While there is a formula that we shall eventually learn for this number, it requires more machinery than we now have available. However, there is a good method for computing this number that is similar to Pascal’s equation.
• 3.3: Partitions of Integers
We have now completed all our distribution problems except for those in which both the objects and the recipients are identical. For example, we might be putting identical apples into identical paper bags. In this case, all that matters is how many bags get one apple, how many get two, how many get three, and so on. Thus, for each bag, we have a number, and the multiset of numbers of apples in the various bags is what determines our distribution of apples into identical bags.
• 3.4: Distribution Problems (Exercises)
This section contains the supplementary problems related to the materials discussed in Chapter 3.

Thumbnail: The 15 partitions of a 4-element set ordered in a Hasse diagram There are S(4,1),...,S(4,4) = 1,7,6,1 partitions containing 1,2,3,4 sets. (CC BY-3.0; Watchduck).