2018-10-27 16:57:32 +02:00
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\documentclass[12pt,a4paper,german]{article}
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\usepackage{url}
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%\usepackage{graphics}
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\usepackage{times}
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\usepackage[T1]{fontenc}
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\usepackage{ngerman}
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\usepackage{float}
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\usepackage{diagbox}
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\usepackage[utf8]{inputenc}
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\usepackage{geometry}
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\usepackage{amsfonts}
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\usepackage{amsmath}
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\usepackage{csquotes}
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\usepackage{graphicx}
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\usepackage{epsfig}
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\usepackage{paralist}
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\geometry{left=2.0cm,textwidth=17cm,top=3.5cm,textheight=23cm}
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%%%%%%%%%% Fill out the the definitions %%%%%%%%%
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\def \name {Valentin Brandl} %
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\def \matrikel {108018274494} %
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\def \pname {Marvin Herrmann} %
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\def \pmatrikel {108018265436} %
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\def \gruppe {2 (Mi 10-12 Andre)}
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\def \uebung {2} %
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% DO NOT MODIFY THIS HEADER
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\newcommand{\hwsol}{
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\vspace*{-2cm}
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\noindent \matrikel \quad \name \hfill \"Ubungsgruppe: \gruppe \\
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\noindent \pmatrikel \quad \pname \\
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\begin{center}{\Large \bf L\"osung f\"ur \"Ubung \# \uebung}\end{center}
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}
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\begin{document}
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%Import header
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\hwsol
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\section*{Aufgabe 2.1}
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\begin{enumerate}[1.]
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\item Kekse $\widehat{=}$ Bälle (unterscheidbar, da \enquote{verschieden}), Portionen $\widehat{=}$ Urnen (nicht
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unterscheidbar). $n = 9$, $k = 5$
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Problem entspricht einer ungeordneten $k$-Mengenpartition, also $S_{n,k}$
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2018-10-28 13:22:52 +01:00
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\begin{tabular}{c|cccccccccc}
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$n/k$ & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\\hline
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0 & 1 &&&&&&&&& \\
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1 & 0 & 1 & & & & & & & & \\
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2 & 0 & 1 & 1 & & & & & & & \\
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3 & 0 &1 &3 &1 & & & & & & \\
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4 & 0 &1 &7 &6 &1 & & & & & \\
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5 & 0 &1 &15 &25 &10 &1 & & & & \\
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6 & 0 &1 &31 &90 &65 &15 &1 & & & \\
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7 & 0 &1 &63 &301 &350 &140 &21 &1 & & \\
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8 & 0 &1 &127 &966 &1701 &1050 &266 &28 &1 & \\
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9 & 0 &1 &255 &3025 &7770 &\underline{6951} &2646 &462 &36 &1 \\
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\end{tabular}
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2018-10-27 16:57:32 +02:00
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\begin{eqnarray*}
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S_{n,k} &=& S_{n-1,k-1} + k * S_{n-1,k} \text{ mit} \\
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S_{0,0} &=& 1 \\
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S_{n,n} &=& 1 \\
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S_{n,1} &=& 1 \\
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S_{n,0} &=& 0 \\\\
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S_{9,5} &=& 6951
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\end{eqnarray*}
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\item Bälle weiterhin unterscheidbar, Urnen jetzt auch unterscheidbar $\Rightarrow$ geordnete Mengenpartition.
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\begin{eqnarray*}
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k! * S_{n.k} &=& 5! * S_{9,5} \\
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&=& 120 * 6951 \\
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&=& 834120
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\end{eqnarray*}
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\item Jetzt gilt Teller $\widehat{=}$ Ball, Keks $\widehat{=}$ Urne. $n = 5$, $k = 3$.
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Urnen sind unterscheidbar, \enquote{fünfgangiges Menü} $\Rightarrow$ Bälle sind auch untescheidbar
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\begin{eqnarray*}
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n^{\underline{k}} &=& 5^{\underline{3}} \\
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&=& 5 * 4 * 3 \\
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&=& 60
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\end{eqnarray*}
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\end{enumerate}
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\end{document}
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