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

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

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

\end{enumerate}

\end{document}

