87 lines
2.7 KiB
TeX
87 lines
2.7 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 \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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\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,2} &=& 2^{n-1} - 1 \\
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S_{n,3} &=& \frac{1}{2}(3^{n-1} - 2^n + 1) \\
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S_{n,0} &=& 0 \\\\
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S_{9,5} &=& S_{8,4} + 5 * S_{8,5} \\
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&=& (S_{7,3} + 4 * S_{7,4}) + 5 * (S_{7,4} + 5 * S_{7,5}) \\
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&=& ((S_{6,2} + 3 * S_{6.3}) + 4 * (S_{6,3} + 4 * S_{6,4})) + 5 * ((S_{6,3} + 4 * S_{6,4}) + 5 *
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(S_{6,4} + 5 * S_{6,5})) \\
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&=& ((32 + 3 * 90) + 4*(90 + 4*(S_{5,3} + 4*S_{5,4}))) + 5 * ((90 + 4*(S_{5,3} + 4*S_{5,4})) \\
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&&+ 5*((S_{5,3}+ 4*S_{5,4}) + 5*(S_{5,4} + 5*S_{5,5}))) \\
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&=& (302 + 4*(90+4*(57 + 4*(S_{4,3} + 4*S_{4,4})))) \\
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&&+ 5*((90+4*(57 + 4*(S_{4,3} + 4*S_{4,4}))) \\
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&&+ 5*((57 + 4*(S_{4,3} + 4*S_{4,4})) + 5 * ((S_{4,3} + 4*S_{4,4}) + 5 * 1))) \\
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&=& (302 + 4*(90 + 4*(57 + 4*(22 + 4*1)))) \\
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&& + 5*((90 + 4*(57 + 4*(22 + 4*1)))) \\
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&& + 5 * ((57 + 4*(22 + 4*1)) + 5*((22 + 4*1) + 5)) \\
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&=& 3238 + 3670 + 1580 \\
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&=& 8488
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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 * 8488 \\
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&=& 1018560
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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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\end{enumerate}
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\end{document}
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