A calculus of the absurd
\(\newcommand{\footnotename}{footnote}\)
\(\def \LWRfootnote {1}\)
\(\newcommand {\footnote }[2][\LWRfootnote ]{{}^{\mathrm {#1}}}\)
\(\newcommand {\footnotemark }[1][\LWRfootnote ]{{}^{\mathrm {#1}}}\)
\(\let \LWRorighspace \hspace \)
\(\renewcommand {\hspace }{\ifstar \LWRorighspace \LWRorighspace }\)
\(\newcommand {\mathnormal }[1]{{#1}}\)
\(\newcommand \ensuremath [1]{#1}\)
\(\newcommand {\LWRframebox }[2][]{\fbox {#2}} \newcommand {\framebox }[1][]{\LWRframebox } \)
\(\newcommand {\setlength }[2]{}\)
\(\newcommand {\addtolength }[2]{}\)
\(\newcommand {\setcounter }[2]{}\)
\(\newcommand {\addtocounter }[2]{}\)
\(\newcommand {\arabic }[1]{}\)
\(\newcommand {\number }[1]{}\)
\(\newcommand {\noalign }[1]{\text {#1}\notag \\}\)
\(\newcommand {\cline }[1]{}\)
\(\newcommand {\directlua }[1]{\text {(directlua)}}\)
\(\newcommand {\luatexdirectlua }[1]{\text {(directlua)}}\)
\(\newcommand {\protect }{}\)
\(\def \LWRabsorbnumber #1 {}\)
\(\def \LWRabsorbquotenumber "#1 {}\)
\(\newcommand {\LWRabsorboption }[1][]{}\)
\(\newcommand {\LWRabsorbtwooptions }[1][]{\LWRabsorboption }\)
\(\def \mathchar {\ifnextchar "\LWRabsorbquotenumber \LWRabsorbnumber }\)
\(\def \mathcode #1={\mathchar }\)
\(\let \delcode \mathcode \)
\(\let \delimiter \mathchar \)
\(\def \oe {\unicode {x0153}}\)
\(\def \OE {\unicode {x0152}}\)
\(\def \ae {\unicode {x00E6}}\)
\(\def \AE {\unicode {x00C6}}\)
\(\def \aa {\unicode {x00E5}}\)
\(\def \AA {\unicode {x00C5}}\)
\(\def \o {\unicode {x00F8}}\)
\(\def \O {\unicode {x00D8}}\)
\(\def \l {\unicode {x0142}}\)
\(\def \L {\unicode {x0141}}\)
\(\def \ss {\unicode {x00DF}}\)
\(\def \SS {\unicode {x1E9E}}\)
\(\def \dag {\unicode {x2020}}\)
\(\def \ddag {\unicode {x2021}}\)
\(\def \P {\unicode {x00B6}}\)
\(\def \copyright {\unicode {x00A9}}\)
\(\def \pounds {\unicode {x00A3}}\)
\(\let \LWRref \ref \)
\(\renewcommand {\ref }{\ifstar \LWRref \LWRref }\)
\( \newcommand {\multicolumn }[3]{#3}\)
\(\require {textcomp}\)
\(\newcommand {\LWRmarginnote }[1][]{}\)
\(\newcommand {\marginnote }[2][]{\qquad {\small \textrm {#2}}\LWRmarginnote }\)
\(\def \LWRsidenote {1}\)
\(\newcommand {\sidenotemark }[1][\LWRsidenote ]{{}^{\mathrm {#1}}}\)
\(\newcommand {\intertext }[1]{\text {#1}\notag \\}\)
\(\let \Hat \hat \)
\(\let \Check \check \)
\(\let \Tilde \tilde \)
\(\let \Acute \acute \)
\(\let \Grave \grave \)
\(\let \Dot \dot \)
\(\let \Ddot \ddot \)
\(\let \Breve \breve \)
\(\let \Bar \bar \)
\(\let \Vec \vec \)
\(\newcommand {\nicefrac }[3][]{\mathinner {{}^{#2}\!/\!_{#3}}}\)
\(\newcommand {\unit }[2][]{#1 \mathinner {#2}}\)
\(\newcommand {\unitfrac }[3][]{#1 \mathinner {{}^{#2}\!/\!_{#3}}}\)
\(\newcommand {\bm }[1]{\boldsymbol {#1}}\)
\(\require {mathtools}\)
\(\newenvironment {crampedsubarray}[1]{}{}\)
\(\newcommand {\smashoperator }[2][]{#2\limits }\)
\(\newcommand {\SwapAboveDisplaySkip }{}\)
\(\newcommand {\LaTeXunderbrace }[1]{\underbrace {#1}}\)
\(\newcommand {\LaTeXoverbrace }[1]{\overbrace {#1}}\)
\(\newcommand {\LWRmultlined }[1][]{\begin {multline*}}\)
\(\newenvironment {multlined}[1][]{\LWRmultlined }{\end {multline*}}\)
\(\let \LWRorigshoveleft \shoveleft \)
\(\renewcommand {\shoveleft }[1][]{\LWRorigshoveleft }\)
\(\let \LWRorigshoveright \shoveright \)
\(\renewcommand {\shoveright }[1][]{\LWRorigshoveright }\)
\(\newcommand {\shortintertext }[1]{\text {#1}\notag \\}\)
\(\newcommand {\vcentcolon }{\mathrel {\unicode {x2236}}}\)
\(\newcommand {\Var }{\operatorname {Var}}\)
\(\newcommand {\Expected }{\operatorname {E}}\)
\(\newcommand {\abs }[1]{\left \lvert #1\right \rvert }\)
\(\newcommand {\norm }[1]{\left \lVert #1\right \rVert }\)
\(\newcommand {\rbrackets }[1]{\left (#1\right )}\)
\(\newcommand {\sbrackets }[1]{\left [#1\right ]}\)
\(\newcommand {\cbrackets }[1]{\left \{#1\right \}}\)
\(\newcommand {\RE }{\operatorname {Re}}\)
\(\newcommand {\IM }{\operatorname {IM}}\)
\(\newcommand {\Span }{\operatorname {span}}\)
16.1.2 From English to maths
One thing that can be tricky in combinatorics is working out what words in English mean in terms of combinatorial operations. Here’s a handy dictionary.
| . |
|---|
|
English
|
Combinatorics
|
|
This can happen in way \(A\) or in way \(B\)
|
number of ways for \(A\) + number of ways for \(B\)
|
|
To have "whatever" I need both \(A\) and then \(B\)
|
number of ways of \(A\) \(\times \) number of ways \(B\)
|
|
For every item in this set of \(n\) objects there are \(k\) ways of obtaining it.
|
\(k^n\)
|
|
I have a group of \(n\) things, and I want to pick \(k\) of them, but I don’t care about the order in which I get them.
|
\(\binom {n}{k}\)
|
|
I have a group of \(n\) different things, and I want to pick \(k\) of them, and I do care about the order in which I get them
|
\(\frac {n!}{(n-k)!}\)
|
One strategy I find very useful in solving combinatorics problems is to write out a description of what I’m after in English, and then translate this into combinatorial operations (e.g. permutations, combinations, etc.). 118118 This strategy works really well throughout mathematics, but it’s especially helpful here.