8: Counting
- Page ID
- 201162
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)"Counting" sounds easy, and sometimes it is. But when there are a large number of things to count, defined by nontrivial rules, the task becomes more challenging. A whole branch of mathematics, called "combinatorics," has evolved to deal with these challenges. This chapter introduces some fundamental ideas from combinatorics and shows how they solve a variety of common counting problems.
- 8.1: Sets and Counting
- A set is a collection of objects. This is a simple idea, but one that applies to almost any collection of any kind of object. Sets are therefore powerful tools for organizing and manipulating data in many settings. This section introduces sets and some related terminology, some operations you can do on sets, and a visual representation of relationships between them.
- 8.2: Tree Diagrams and the Multiplication Axiom
- A common kind of counting problem revolves around the outcomes of a multi-step process, with some number of ways to complete the first step, followed by another number of ways to complete the second, and so forth. This section presents a graphical way to represent such processes and their outcomes, which in turn inspires a formulaic way to count the outcomes.
- 8.3: Permutations
- Another important class of counting problems involves arranging items in some order. Such arrangements are called "permutations" of the items. Counting permutations is both important in its own right, and a valuable tool for counting more complicated arrangements. This section defines the notion of permutation, and develops related counting formulas.
- 8.4: Circular Permutations and Permutations with Similar Elements
- The motivation for permutations is arrangements of distinct items in a straight line. But what if the items aren't distinct, or they're arranged in a circle with no clear notion of a first element? This section develops math for counting the number of possible arrangements under these conditions.
- 8.5: Combinations
- Permutations and their variations are arrangements of items in which order matters: two arrangements of the same items but in a different order are different permutations. It's also possible to arrange (or maybe "select" is a better word now) items without concern for their order. Such selections are called "combinations." This section explores the difference between combinations and permutations, and develops math for counting combinations.
- 8.6: Combinations Involving Several Sets
- Counting problems that involve collecting items from several sets often break into subproblems solvable with the combinations and permutations we've already studied. This section presents some examples of such problems and of how to combine solutions to the subproblems into a solution to the whole.
- 8.7: Binomial Theorem
- Combinations are elegantly but perhaps unexpectedly useful for calculating powers of binomials. So useful, in fact, that combinations and powers of binomials are the subject of an important theorem in algebra, the Binomial Theorem. This section introduces that theorem and explains why it holds.