85 lines
2.6 KiB
TeX
85 lines
2.6 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 2.3}
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\begin{enumerate}[1.]
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\item
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\begin{eqnarray*}
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x_1 + x_2 + x_3 + x_4 + x_5 + x_6 &=& 67 \text{ mit } x_i \ge 0 \text{ für } 1 \le i \le 6 \\
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\text{Normalisierung:} \\
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\text{für } 1 \le i \le 3 \rightarrow x_i' &=& x_i - 1 \text{ (ungerade Zahlen werden gerade)} \\
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\text{für } 4 \le i \le 6 \rightarrow x_i' &=& x_i \\
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\Rightarrow x_1' + x_2' + x_3' + x_4' + x_5' + x_6' &=& 64 \\
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\text{für } 1 \le i \le 6 \rightarrow y_i &=& \frac{x_i'}{2}
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\text{ (Bedingung \enquote{alle Zahlen gerade} erfüllt)} \\
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\Rightarrow y_1 + y_2 + y_3 + y_4 + y_5 + y_6 &=& 32 \\
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\text{für } 1 \le i \le 6 \rightarrow z_i &=& y_i + 1 \\
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\Rightarrow z_1 + z_2 + z_3 + z_4 + z_5 + y_6 &=& 38 \\\\
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\Rightarrow \binom{n-1}{k-1} &=& \binom{37}{5} = 435897
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\end{eqnarray*}
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\item
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\begin{tabular}{c|cccccccccc}
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$n/k$ & 1 & 2 & 3 & 4 & 5 &6 & 7 & 8 & 9 & 10 \\\hline
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1 & 1 & & & & & & & & & \\
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2 & 1 & 1 & & & & & & & & \\
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3 & 1 & 1 & 1 & & & & & & & \\
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4 & 1 & 2 & 1 & 1 & & & & & & \\
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5 & 1 & 2 & 2 & 1 & 1 & & & & & \\
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6 & 1 & 3 & 3 & 2 & 1 & 1 & & & & \\
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7 & 1 & 3 & 4 & 3 & 2 & 1 & 1 & & & \\
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8 & 1 & 4 & 5 & 5 & 3 & 2 & 1 & 1 & & \\
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9 & 1 & 4 & 7 & 6 & 5 & 3 & 2 & 1 & 1 & \\
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10 & 1 & 5 & 8 & \underline{9} &7 &5 &3 &2 & 1 & 1 \\
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\end{tabular}
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\begin{eqnarray*}
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P_{n,k} &=& P_{n-1,k-1} + P_{n-k,k} \text{ mit} \\
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P_{n.0} &=& 0 \\
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P_{n.n} &=& 1 \\
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P_{n.1} &=& 1 \\\\
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P_{10,4} &=& 9
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\end{eqnarray*}
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\end{enumerate}
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\end{document}
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