What This Document Is
This document comprises the eighth set of lecture notes for Hunter College CUNY’s Precalculus course (MATH 125). It introduces fundamental concepts in combinatorics – the mathematics of counting – and builds upon previously established ideas regarding sets and ordered pairs. The notes specifically cover lists (ordered sequences), factorials, and counting subsets. It draws definitions and examples from Richard Hammack’s “Book of Proof.”
Why This Document Matters
These notes are essential for students enrolled in MATH 125 who need a detailed record of the lecture material on counting techniques. Understanding these concepts is crucial for success in precalculus and forms a foundation for more advanced mathematical topics like probability and statistics. This material is typically covered early in a combinatorics unit, providing the tools to solve a wide range of counting problems.
Common Limitations or Challenges
This document provides a foundational overview and does *not* offer practice problems with solutions, nor does it delve into complex applications of these counting principles. It’s a record of concepts, not a complete self-study guide. Students will still need to engage with textbook exercises, homework assignments, and potentially seek additional resources to fully master the material.
What This Document Provides
The full document includes:
* A formal definition of “lists” and their properties (length, entries, equality).
* Illustrative examples of lists in real-world contexts (phone numbers, zip codes).
* The Multiplication Principle (Fact 3.1) for counting lists.
* The definition of factorials and their application to non-repetitive lists.
* A formula for calculating the number of non-repetitive lists chosen from a larger set.
* An introduction to counting subsets.
* Examples illustrating how to apply these concepts to solve basic counting problems.
This preview *does not* include detailed solutions to example problems, nor does it cover more advanced counting techniques beyond the basics presented in sections 3.1, 3.2, and 3.3. It also does not include any practice exercises.