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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 2.1}
\begin{enumerate}[1.]
	\item Kekse $\widehat{=}$ Bälle (unterscheidbar, da \enquote{verschieden}), Portionen $\widehat{=}$ Urnen (nicht
		unterscheidbar). $n = 9$, $k = 5$

		Problem entspricht einer ungeordneten $k$-Mengenpartition, also $S_{n,k}$

		\begin{eqnarray*}
			S_{n,k} &=& S_{n-1,k-1} + k * S_{n-1,k} \text{ mit} \\
			S_{0,0} &=& 1 \\
			S_{n,n} &=& 1 \\
			S_{n,1} &=& 1 \\
			S_{n,2} &=& 2^{n-1} - 1 \\
			S_{n,3} &=& \frac{1}{2}(3^{n-1} - 2^n + 1) \\
			S_{n,0} &=& 0 \\\\
			S_{9,5} &=& S_{8,4} + 5 * S_{8,5} \\
					&=& (S_{7,3} + 4 * S_{7,4}) + 5 * (S_{7,4} + 5 * S_{7,5}) \\
					&=& ((S_{6,2} + 3 * S_{6.3}) + 4 * (S_{6,3} + 4 * S_{6,4})) + 5 * ((S_{6,3} + 4 * S_{6,4}) + 5 *
			(S_{6,4} + 5 * S_{6,5})) \\
			&=& ((32 + 3 * 90) + 4*(90 + 4*(S_{5,3} + 4*S_{5,4}))) + 5 * ((90 + 4*(S_{5,3} + 4*S_{5,4})) \\
			&&+ 5*((S_{5,3}+ 4*S_{5,4}) + 5*(S_{5,4} + 5*S_{5,5}))) \\
			&=& (302 + 4*(90+4*(57 + 4*(S_{4,3} + 4*S_{4,4})))) \\
			&&+ 5*((90+4*(57 + 4*(S_{4,3} + 4*S_{4,4}))) \\
			&&+ 5*((57 + 4*(S_{4,3} + 4*S_{4,4})) + 5 * ((S_{4,3} + 4*S_{4,4}) + 5 * 1))) \\
			&=& (302 + 4*(90 + 4*(57 + 4*(22 + 4*1)))) \\
			&& + 5*((90 + 4*(57 + 4*(22 + 4*1)))) \\
			&& + 5 * ((57 + 4*(22 + 4*1)) + 5*((22 + 4*1) + 5)) \\
			&=& 3238 + 3670 + 1580 \\
			&=& 8488
		\end{eqnarray*}

	\item Bälle weiterhin unterscheidbar, Urnen jetzt auch unterscheidbar $\Rightarrow$ geordnete Mengenpartition.

		\begin{eqnarray*}
			k! * S_{n.k} &=& 5! * S_{9,5} \\
						 &=& 120 * 8488 \\
						 &=& 1018560
		\end{eqnarray*}

	\item Jetzt gilt Teller $\widehat{=}$ Ball, Keks $\widehat{=}$ Urne. $n = 5$, $k = 3$.

		Urnen sind unterscheidbar, \enquote{fünfgangiges Menü} $\Rightarrow$ Bälle sind auch untescheidbar

\end{enumerate}

\end{document}