\relax 
\@writefile{toc}{\contentsline {chapter}{\numberline {3}Combinatorics}{75}}
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\newlabel{chp 3}{{3}{75}}
\@writefile{toc}{\contentsline {section}{\numberline {3.1}Permutations}{75}}
\newlabel{sec 3.1}{{3.1}{75}}
\newlabel{exam 3.1}{{3.1}{75}}
\@writefile{lof}{\contentsline {figure}{\numberline {3.1}{\ignorespaces Tree for your menu.}}{76}}
\newlabel{fig 3.1}{{3.1}{76}}
\@writefile{lof}{\contentsline {figure}{\numberline {3.2}{\ignorespaces Two-stage probability assignment.}}{77}}
\newlabel{fig 3.2}{{3.2}{77}}
\newlabel{exam 3.2}{{3.2}{77}}
\newlabel{exam 3.3}{{3.3}{77}}
\@writefile{lot}{\contentsline {table}{\numberline {3.1}{\ignorespaces Birthday problem.}}{78}}
\newlabel{table 3.1}{{3.1}{78}}
\@writefile{lot}{\contentsline {table}{\numberline {3.2}{\ignorespaces Birthday problem.}}{79}}
\newlabel{table 3.2}{{3.2}{79}}
\newlabel{def 3.1}{{3.1}{79}}
\@writefile{lot}{\contentsline {table}{\numberline {3.3}{\ignorespaces Values of the factorial function.}}{80}}
\newlabel{table 3.25}{{3.3}{80}}
\newlabel{thm 3.1}{{3.1}{80}}
\newlabel{thm 3.1.1}{{3.1}{80}}
\newlabel{thm 3.2}{{3.2}{80}}
\newlabel{def 3.2}{{3.3}{80}}
\@writefile{lot}{\contentsline {table}{\numberline {3.4}{\ignorespaces Stirling approximations to the factorial function.}}{81}}
\newlabel{table 3.26}{{3.4}{81}}
\newlabel{exam 3.4}{{3.4}{81}}
\newlabel{thm 3.3}{{3.3}{81}}
\@writefile{lot}{\contentsline {table}{\numberline {3.5}{\ignorespaces Fixed point distributions.}}{82}}
\newlabel{table 3.3}{{3.5}{82}}
\@writefile{lot}{\contentsline {table}{\numberline {3.6}{\ignorespaces Snowfall in Hanover.}}{83}}
\newlabel{table 3.4}{{3.6}{83}}
\@writefile{lot}{\contentsline {table}{\numberline {3.7}{\ignorespaces Ranking of total snowfall.}}{83}}
\newlabel{table 3.5}{{3.7}{83}}
\newlabel{def 3.3}{{3.4}{83}}
\newlabel{exer 3.1.1}{{1}{88}}
\newlabel{exer 3.1.2}{{2}{88}}
\newlabel{exer 3.1.3}{{3}{88}}
\newlabel{exer 3.1.4}{{4}{88}}
\newlabel{exer 3.1.5}{{5}{88}}
\newlabel{exer 3.1.6}{{6}{88}}
\newlabel{exer 3.1.7}{{7}{89}}
\newlabel{exer 3.1.8}{{8}{89}}
\newlabel{exer 3.1.9}{{9}{89}}
\newlabel{exer 3.1.10}{{10}{89}}
\newlabel{exer 3.1.11}{{11}{89}}
\newlabel{exer 3.1.12}{{12}{89}}
\newlabel{exer 3.1.13}{{13}{89}}
\newlabel{exer 3.1.14}{{14}{89}}
\newlabel{exer 3.1.15}{{15}{90}}
\newlabel{exer 3.1.16}{{16}{90}}
\newlabel{exer 3.1.17}{{17}{90}}
\newlabel{exer 3.1.18}{{18}{90}}
\newlabel{exer 3.1.19}{{19}{90}}
\newlabel{exer 3.1.20}{{20}{91}}
\newlabel{exer 3.1.21}{{21}{91}}
\newlabel{exer 3.1.22}{{22}{91}}
\newlabel{exer 3.1.23}{{23}{92}}
\newlabel{exer 3.1.24}{{24}{92}}
\@writefile{toc}{\contentsline {section}{\numberline {3.2}Combinations}{92}}
\newlabel{sec 3.2}{{3.2}{92}}
\newlabel{exam 3.7}{{3.5}{93}}
\newlabel{thm 3.6}{{3.4}{93}}
\newlabel{eq 3.3}{{3.1}{93}}
\@writefile{lof}{\contentsline {figure}{\numberline {3.3}{\ignorespaces Pascal's triangle.}}{94}}
\newlabel{fig 3.6}{{3.3}{94}}
\newlabel{thm 3.7}{{3.5}{94}}
\newlabel{eq 3.4}{{3.2}{94}}
\newlabel{exam 3.8}{{3.6}{95}}
\@writefile{lof}{\contentsline {figure}{\numberline {3.4}{\ignorespaces Tree diagram of three Bernoulli trials.}}{96}}
\newlabel{fig 3.7}{{3.4}{96}}
\newlabel{def 3.4}{{3.5}{96}}
\newlabel{exam 3.9}{{3.7}{96}}
\newlabel{thm 3.8}{{3.6}{98}}
\newlabel{exam 3.10}{{3.8}{98}}
\newlabel{exam 3.11}{{3.9}{98}}
\@writefile{lot}{\contentsline {table}{\numberline {3.8}{\ignorespaces Binomial probabilities for $n = 100,\ p = 1/2$.}}{99}}
\newlabel{table 3.27}{{3.8}{99}}
\newlabel{def 3.5}{{3.6}{99}}
\newlabel{exam 3.2.1}{{3.10}{99}}
\@writefile{lof}{\contentsline {figure}{\numberline {3.5}{\ignorespaces Binomial distributions.}}{100}}
\newlabel{fig 3.8}{{3.5}{100}}
\@writefile{lof}{\contentsline {figure}{\numberline {3.6}{\ignorespaces Simulation of the Galton board.}}{101}}
\newlabel{fig 2.22}{{3.6}{101}}
\newlabel{exam 3.12}{{3.11}{101}}
\@writefile{lof}{\contentsline {figure}{\numberline {3.7}{\ignorespaces The power curve.}}{103}}
\newlabel{fig 3.9}{{3.7}{103}}
\newlabel{thm 3.9}{{3.7}{103}}
\newlabel{cor 3.1}{{3.1}{103}}
\newlabel{thm 3.10}{{3.8}{104}}
\newlabel{eq 3.5}{{3.3}{104}}
\newlabel{exam 3.13}{{3.12}{105}}
\@writefile{lot}{\contentsline {table}{\numberline {3.9}{\ignorespaces Hat check problem.}}{106}}
\newlabel{table 3.7}{{3.9}{106}}
\newlabel{exam 3.14}{{3.13}{107}}
\newlabel{exam 3.15}{{3.14}{107}}
\@writefile{lot}{\contentsline {table}{\numberline {3.10}{\ignorespaces Pascal's triangle.}}{108}}
\newlabel{table 3.8}{{3.10}{108}}
\@writefile{lot}{\contentsline {table}{\numberline {3.11}{\ignorespaces Figurate numbers.}}{108}}
\newlabel{table 3.9}{{3.11}{108}}
\@writefile{lof}{\contentsline {figure}{\numberline {3.8}{\ignorespaces Pythagorean triangular patterns.}}{109}}
\newlabel{fig 3.10}{{3.8}{109}}
\@writefile{lof}{\contentsline {figure}{\numberline {3.9}{\ignorespaces Geometric representation of the tetrahedral number 10.}}{109}}
\newlabel{fig 3.11}{{3.9}{109}}
\@writefile{lot}{\contentsline {table}{\numberline {3.12}{\ignorespaces Outcomes for the roll of two dice.}}{110}}
\newlabel{table 3.10}{{3.12}{110}}
\@writefile{lof}{\contentsline {figure}{\numberline {3.10}{\ignorespaces Chu Shih-chieh's triangle. [From J.\ Needham, {\em  {Science and Civilization in China,}\/} vol. 3 (New York: Cambridge University Press, 1959), p. 135. Reprinted with permission.]}}{111}}
\newlabel{fig 3.12}{{3.10}{111}}
\@writefile{lot}{\contentsline {table}{\numberline {3.13}{\ignorespaces Pascal's solution for the problem of points.}}{113}}
\newlabel{table 3.11}{{3.13}{113}}
\newlabel{exer 3.2.1}{{1}{113}}
\newlabel{exer 3.2.2}{{2}{113}}
\newlabel{exer 3.2.3}{{3}{113}}
\newlabel{exer 3.2.4}{{4}{113}}
\newlabel{exer 3.2.5}{{5}{114}}
\newlabel{exer 3.2.6}{{6}{114}}
\newlabel{exer 3.2.7}{{7}{114}}
\newlabel{exer 3.2.8}{{8}{114}}
\newlabel{exer 3.2.9}{{9}{114}}
\newlabel{exer 3.2.10}{{10}{114}}
\newlabel{exer 3.2.11}{{11}{114}}
\newlabel{exer 3.2.12}{{12}{114}}
\newlabel{exer 3.2.13}{{13}{114}}
\newlabel{exer 3.2.14}{{14}{115}}
\newlabel{exer 3.2.15}{{15}{115}}
\newlabel{exer 3.2.16}{{16}{115}}
\newlabel{exer 3.2.17}{{17}{115}}
\newlabel{exer 3.2.18}{{18}{115}}
\newlabel{exer 3.2.19}{{19}{115}}
\newlabel{exer 3.2.20}{{20}{115}}
\newlabel{exer 3.2.21}{{21}{115}}
\newlabel{exer 3.2.22}{{22}{116}}
\newlabel{exer 3.2.23}{{23}{116}}
\newlabel{exer 3.2.24}{{24}{116}}
\newlabel{exer 3.2.25}{{25}{116}}
\newlabel{exer 3.2.26}{{26}{116}}
\newlabel{exer 3.2.27}{{27}{116}}
\newlabel{exer 3.2.28}{{28}{117}}
\newlabel{exer 3.2.29}{{29}{117}}
\newlabel{exer 3.2.30}{{30}{117}}
\newlabel{exer 3.2.31}{{31}{117}}
\newlabel{exer 3.2.32}{{32}{117}}
\newlabel{exer 3.2.33}{{33}{117}}
\newlabel{exer 3.2.34}{{34}{118}}
\newlabel{exer 3.2.35}{{35}{118}}
\newlabel{exer 3.2.35.5}{{36}{118}}
\newlabel{exer 3.2.36}{{37}{118}}
\newlabel{exer 3.2.37}{{38}{119}}
\newlabel{exer 3.2.38}{{39}{119}}
\newlabel{exer 3.2.39}{{40}{119}}
\@writefile{toc}{\contentsline {section}{\numberline {3.3}Card Shuffling}{120}}
\newlabel{sec 3.3}{{3.3}{120}}
\newlabel{thm 3.3.1}{{3.9}{122}}
\newlabel{thm 3.3.2}{{3.10}{123}}
\@writefile{lof}{\contentsline {figure}{\numberline {3.11}{\ignorespaces Before a 2-unshuffle.}}{124}}
\newlabel{fig 3.14}{{3.11}{124}}
\@writefile{lof}{\contentsline {figure}{\numberline {3.12}{\ignorespaces Before a 4-unshuffle.}}{124}}
\newlabel{fig 3.15}{{3.12}{124}}
\newlabel{eq 3.3.1}{{3.4}{126}}
\newlabel{thm 3.3.3}{{3.11}{126}}
\newlabel{thm 3.3.4}{{3.12}{127}}
\newlabel{eq 3.6}{{3.5}{127}}
\@writefile{lot}{\contentsline {table}{\numberline {3.14}{\ignorespaces Distance to the random process.}}{129}}
\newlabel{table 3.12}{{3.14}{129}}
\@writefile{lof}{\contentsline {figure}{\numberline {3.13}{\ignorespaces Distance to the random process.}}{129}}
\newlabel{fig 3.13}{{3.13}{129}}
\newlabel{exer 3.3.1}{{1}{131}}
\newlabel{exer 3.3.2}{{2}{131}}
\newlabel{exer 3.3.3}{{3}{131}}
\newlabel{exer 3.3.4}{{4}{132}}
\newlabel{exer 3.3.5}{{5}{132}}
\@setckpt{ch3}{
\setcounter{page}{133}
\setcounter{equation}{5}
\setcounter{enumi}{5}
\setcounter{enumii}{2}
\setcounter{enumiii}{0}
\setcounter{enumiv}{0}
\setcounter{footnote}{29}
\setcounter{mpfootnote}{0}
\setcounter{part}{0}
\setcounter{chapter}{3}
\setcounter{section}{3}
\setcounter{subsection}{0}
\setcounter{subsubsection}{0}
\setcounter{paragraph}{0}
\setcounter{subparagraph}{0}
\setcounter{figure}{13}
\setcounter{table}{14}
\setcounter{examp}{14}
\setcounter{defn}{6}
\setcounter{thm}{12}
\setcounter{cor}{1}
\setcounter{prop}{0}
\setcounter{lem}{0}
}