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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 \qname {Pascal Brackmann}
\def \qmatrikel {108017113834}				%
\def \uebung {3}								%
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\newcommand{\hwsol}{
\vspace*{-2cm}
\noindent \matrikel \quad \name \hfill \"Ubungsgruppe: \gruppe \\
\noindent \pmatrikel \quad \pname \\
\noindent \qmatrikel \quad \qname \\
\begin{center}{\Large \bf L\"osung f\"ur \"Ubung \# \uebung}\end{center}
}

\begin{document}
%Import header
\hwsol

\section*{Aufgabe 3.1}
\begin{enumerate}[1.]

	\item Getränke $\widehat{=}$ Bälle (unterscheidbar); Tablets $\widehat{=}$ Urnen (nicht unterscheidbar)

		In jeder Urne mindestens ein Ball $\Rightarrow$ surjektiv.

		$\Rightarrow$ $S_{n,m}$ mit $n = 8$, $m = 4$

		\begin{tabular}{c|cccccccccc}
			$n/m$ & 0 & 1 & 2 & 3 & 4 \\\hline
			0 & 1 &&&&\\
			1 & 0 & 1 & &  & \\
			2 & 0 & 1 & 1  & & \\
			3 & 0 &1 &3 &1  & \\
			4 & 0 &1 &7 &6 &1  \\
			5 & 0 &1 &15 &25 &10 \\
			6 & 0 &1 &31 &90 &65 \\
			7 & 0 &1 &63 &301 &350 \\
			8 & 0 &1 &127 &966 &\underline{1701} \\
		\end{tabular}

		\begin{eqnarray*}
			S_{8,4} = 1701
		\end{eqnarray*}

	\item
		\begin{align}
			S_{8,4} &+& S_{8,3} &+& S_{8,2} &+& S_{8,1} &=& \\
			1701 &+& 966 &+& 127 &+& 1 &=& 2795
		\end{align}

	\item Freunde $\widehat{=}$ Urnen (unterscheidbar); Getränke $\widehat{=}$ Bällen (unterscheidbar).

		Mindestens ein Ball pro Urne $\Rightarrow$ surjektiv.

		$\Rightarrow$ $m! * S_{n,m}$ mit $n = 8$, $m = 4$.

		\begin{eqnarray*}
			4! * S_{8,4} &= 24 * 1701 \\
						 &= 40824
		\end{eqnarray*}

\end{enumerate}

\end{document}