3: Sets and Counting
- Page ID
- 216171
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)In this chapter, you will learn to:
- Use set theory and Venn diagrams to solve counting problems.
- Use the Multiplication Axiom to solve counting problems.
- Use Permutations to solve counting problems.
- Use Combinations to solve counting problems.
- Use the Binomial Theorem to expand \((x+y)^n\)
- 3.1: Sets and Counting
- A set is a collection of objects, and its members are called the elements of the set. We name the set by using capital letters, and enclose its members in braces.
- 3.2: Tree Diagrams and the Multiplication Axiom
- This page covers essential counting techniques for probability, emphasizing the Multiplication Axiom and the use of tree diagrams to visualize outcomes in multi-step scenarios. It illustrates how to calculate combinations efficiently using examples like selecting outfits, forming license plates, answering true-false tests, and arranging people. The Multiplication Axiom states that if a task can be done in \(m\) ways and another in \(n\) ways, both tasks can be done in \(m \cdot n\) ways.
- 3.3: Permutations
- This page introduces permutations as ordered arrangements of elements where order matters. It defines factorials and explains how to calculate permutations of n items taken r at a time, offering two key formulas. The multiplication axiom is utilized, with examples illustrating applications in finding arrangements and specific position conditions. Overall, this page establishes foundational concepts in permutations and factorials essential for combinatorial mathematics.
- 3.4: Circular Permutations and Permutations with Similar Elements
- This page explains how to count permutations in circular and repeated item contexts. It details that circular permutations for \(n\) items equal \((n-1)!\), treating rotations as identical. It also discusses permutations with identical items, using "MISSISSIPPI" as an example and the formula \(\frac{n!}{r_{1}! r_{2}! \ldots r_{k}!}\). Practical examples help readers understand and apply these methods effectively.
- 3.5: Combinations
- This page explains the difference between permutations and combinations, highlighting that permutations are order-sensitive arrangements, while combinations are order-insensitive selections. The relationship is illustrated through the formula \(C(n, r) = \frac{P(n, r)}{r!}\). Practical examples, such as committee formations and route selections, underscore the core concepts and calculations of combinations.
- 3.6: Combinations - Involving Several Sets
- This page discusses advanced counting techniques in combinatorics, emphasizing selection from multiple sets with restrictions. It details how to compute combinations for varied scenarios, like forming committees based on gender or grade levels. Examples include creating word sequences from letters and drawing specific 5-card hands from a deck, illustrating the use of the multiplication axiom in combinatorial problems.
- 3.7: Binomial Theorem
- This page introduces combinations through the Binomial Theorem, explaining how to expand expressions like \((x + y)^n\) using coefficients without manual multiplication. It provides examples for expanding \((x+y)^3\) and determining specific coefficients, leading to the general Binomial Theorem. Further examples include expanding more complex terms like \((3a-2b)^4\) and identifying specific terms in those expansions.
- 3.8: Review of Counting
- This page explores combinatorics and probability through various problems, focusing on counting methods, arrangements, selections, and combinations. It includes examples from class enrollment, reading habits, and sports positions, demonstrating calculations of possible outcomes such as team formations and object arrangements.