90 lines
2.2 KiB
TeX
90 lines
2.2 KiB
TeX
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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 \qname {Pascal Brackmann}
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\def \qmatrikel {108017113834} %
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\def \uebung {3} %
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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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\noindent \qmatrikel \quad \qname \\
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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 3.1}
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\begin{enumerate}[1.]
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\item Getränke $\widehat{=}$ Bälle (unterscheidbar); Tablets $\widehat{=}$ Urnen (nicht unterscheidbar)
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In jeder Urne mindestens ein Ball $\Rightarrow$ surjektiv.
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$\Rightarrow$ $S_{n,m}$ mit $n = 8$, $m = 4$
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\begin{tabular}{c|cccccccccc}
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$n/m$ & 0 & 1 & 2 & 3 & 4 \\\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 \\
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6 & 0 &1 &31 &90 &65 \\
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7 & 0 &1 &63 &301 &350 \\
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8 & 0 &1 &127 &966 &\underline{1701} \\
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\end{tabular}
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\begin{eqnarray*}
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S_{8,4} = 1701
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\end{eqnarray*}
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\item
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\begin{align}
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S_{8,4} &+& S_{8,3} &+& S_{8,2} &+& S_{8,1} &=& \\
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1701 &+& 966 &+& 127 &+& 1 &=& 2795
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\end{align}
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\item Freunde $\widehat{=}$ Urnen (unterscheidbar); Getränke $\widehat{=}$ Bällen (unterscheidbar).
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Mindestens ein Ball pro Urne $\Rightarrow$ surjektiv.
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$\Rightarrow$ $m! * S_{n,m}$ mit $n = 8$, $m = 4$.
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\begin{eqnarray*}
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4! * S_{8,4} &= 24 * 1701 \\
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&= 40824
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\end{eqnarray*}
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\end{enumerate}
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\end{document}
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