2 Counting Problems

# 2 Counting Problems - Math 100 Partitioning Problem Martin...

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Unformatted text preview: Math 100 Partitioning Problem Martin H. Weissman 1. A first problem In this problem, students will be asked to answer the following question: Suppose that n is a positive integer. Let S be a set with 3 n elements. How many ways are there to divide S into three teams of n ? In other words, how many sets { A,B,C } are there, such that A,B,C are all subsets of S with n elements, A B = , B C = , and C A = ? To get started, we observe that the following choice-sequences are equivalent: First, divide S into three teams of n elements. Then, label the three teams first, second, and third. First, choose a subset T 1 of S with n elements and call it the first team. Then, choose a subset T 2 of S- T 1 with n elements, and call it the second team. Call the remaining elements of S the third team. We leave it to the students (in pairs and triples) to solve the problem from here....
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