<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\documentclass[12pt,a4paper,german]{article}
\usepackage{url}
%\usepackage{graphics}
\usepackage{times}
\usepackage[T1]{fontenc}
\usepackage{ngerman}
\usepackage{float}
\usepackage{diagbox}
\usepackage[utf8]{inputenc}
\usepackage{geometry}
\usepackage{amsfonts}
\usepackage{amsmath}
\usepackage{csquotes}
\usepackage{graphicx}
\usepackage{epsfig}
\usepackage{paralist}
\geometry{left=2.0cm,textwidth=17cm,top=3.5cm,textheight=23cm}

%%%%%%%%%% Fill out the the definitions %%%%%%%%%
\def \name {Valentin Brandl}					%
\def \matrikel {108018274494}				%
\def \pname {Marvin Herrmann}				%
\def \pmatrikel {108018265436}				%
\def \gruppe {2 (Mi 10-12 Andre)}
\def \uebung {2}								%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 % DO NOT MODIFY THIS HEADER
\newcommand{\hwsol}{
\vspace*{-2cm}
\noindent \matrikel \quad \name \hfill \"Ubungsgruppe: \gruppe \\
\noindent \pmatrikel \quad \pname \\
\begin{center}{\Large \bf L\"osung f\"ur \"Ubung \# \uebung}\end{center}
}

\begin{document}
%Import header
\hwsol

\section*{Aufgabe 2.3}
\begin{enumerate}[1.]

	\item
		\begin{eqnarray*}
			x_1 + x_2 + x_3 + x_4 + x_5 + x_6 &amp;=&amp; 67 \text{ mit } x_i \ge 0 \text{ für } 1 \le i \le 6 \\
			\text{Normalisierung:} \\
			\text{für } 1 \le i \le 3 \rightarrow x_i' &amp;=&amp; x_i - 1 \text{ (ungerade Zahlen werden gerade)} \\
			\text{für } 4 \le i \le 6 \rightarrow x_i' &amp;=&amp; x_i \\
			\Rightarrow x_1' + x_2' + x_3' + x_4' + x_5' + x_6' &amp;=&amp; 64 \\
			\text{für } 1 \le i \le 6 \rightarrow y_i &amp;=&amp; \frac{x_i'}{2}
				\text{ (Bedingung \enquote{alle Zahlen gerade} erfüllt)} \\
			\Rightarrow y_1 + y_2 + y_3 + y_4 + y_5 + y_6 &amp;=&amp; 32 \\
			\text{für } 1 \le i \le 6 \rightarrow z_i &amp;=&amp; y_i + 1 \\
			\Rightarrow z_1 + z_2 + z_3 + z_4 + z_5 + y_6 &amp;=&amp; 38 \\\\
			\Rightarrow \binom{n-1}{k-1} &amp;=&amp; \binom{37}{5} = 435897
		\end{eqnarray*}

	\item
		\begin{tabular}{c|cccccccccc}
			$n/k$ &amp; 1 &amp; 2 &amp; 3 &amp; 4 &amp; 5 &amp;6 &amp; 7 &amp; 8 &amp; 9 &amp; 10 \\\hline
			1 &amp; 1 &amp;   &amp;   &amp;   &amp;   &amp;   &amp;   &amp;   &amp;   &amp;    \\
			2 &amp; 1 &amp; 1 &amp;   &amp;   &amp;   &amp;   &amp;   &amp;   &amp;   &amp;    \\
			3 &amp; 1 &amp; 1 &amp; 1 &amp;   &amp;   &amp;   &amp;   &amp;   &amp;   &amp;    \\
			4 &amp; 1 &amp; 2 &amp; 1 &amp; 1 &amp;   &amp;   &amp;   &amp;   &amp;   &amp;    \\
			5 &amp; 1 &amp; 2 &amp; 2 &amp; 1 &amp; 1 &amp;   &amp;   &amp;   &amp;   &amp;    \\
			6 &amp; 1 &amp; 3 &amp; 3 &amp; 2 &amp; 1 &amp; 1 &amp;   &amp;   &amp;   &amp;    \\
			7 &amp; 1 &amp; 3 &amp; 4 &amp; 3 &amp; 2 &amp; 1 &amp; 1 &amp;   &amp;   &amp;    \\
			8 &amp; 1 &amp; 4 &amp; 5 &amp; 5 &amp; 3 &amp; 2 &amp; 1 &amp; 1 &amp;   &amp;    \\
			9 &amp; 1 &amp; 4 &amp; 7 &amp; 6 &amp; 5 &amp; 3 &amp; 2 &amp; 1 &amp; 1 &amp;    \\
			10 &amp; 1 &amp; 5 &amp; 8 &amp; \underline{9} &amp;7  &amp;5  &amp;3  &amp;2  &amp; 1 &amp; 1  \\
		\end{tabular}

		\begin{eqnarray*}
			P_{n,k} &amp;=&amp; P_{n-1,k-1} + P_{n-k,k} \text{ mit} \\
			P_{n.0} &amp;=&amp; 0 \\
			P_{n.n} &amp;=&amp; 1 \\
			P_{n.1} &amp;=&amp; 1 \\\\
			P_{10,4} &amp;=&amp; 9
		\end{eqnarray*}

\end{enumerate}

\end{document}

</pre></body></html>