Introduction
It has been estimated that the average adult makes 35,000 decisions per day (7). Some of these decisions are only two options, being ‘yes’ or ‘no’. Other decisions are more complicated and may involve breaking a situation down into smaller choices. When was the last time you were in a situation where you thought to yourself, “There are too many options, I can’t decide what I want!” or “What are the chances?” Nearly all situations can be quantified using methods or concepts in combinatorics, the branch of mathematics that includes selections and arrangements of objects with prescribed conditions and configurations. Events are the set of all possible outcomes resulting from a particular situation, experiment, or process. People use basic principles of combinatorics in many everyday situations. Counting allows us to enumerate all possible ways an event can occur, and from these counts we can make inferences that will inform our decision making.
The Fundamental Counting Principle is introduced in elementary and middle school and forms the foundation for enumerating quantities given varying choices. In high school, permutations and combinations are emphasized in Integrated Math II (or Algebra II) and the Math Analysis (precalculus) courses. The connections between these three topics is often not explained or emphasized, though the structures and formulas for computation are similar. This unit emphasizes the relationships between the three key content concepts and gives students the opportunity to understand the general formulas and properties through relatable examples. Furthermore the connection between the number of combinations and the coefficients of the binomial theorem will be demonstrated through examples.
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