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Split exercises

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Valentin Brandl 2018-10-28 13:22:52 +01:00
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73
school/di-ma/uebung/02/02_1.tex
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@ -44,6 +44,20 @@
Problem entspricht einer ungeordneten $k$-Mengenpartition, also $S_{n,k}$
\begin{tabular}{c|cccccccccc}
$n/k$ & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\\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 &1 & & & & \\
6 & 0 &1 &31 &90 &65 &15 &1 & & & \\
7 & 0 &1 &63 &301 &350 &140 &21 &1 & & \\
8 & 0 &1 &127 &966 &1701 &1050 &266 &28 &1 & \\
9 & 0 &1 &255 &3025 &7770 &\underline{6951} &2646 &462 &36 &1 \\
\end{tabular}
\begin{eqnarray*}
S_{n,k} &=& S_{n-1,k-1} + k * S_{n-1,k} \text{ mit} \\
S_{0,0} &=& 1 \\
@ -73,64 +87,5 @@
\end{enumerate}
\section*{Aufgabe 2.2}
\begin{enumerate}[1.]
\item Zyklenzerlegung: $(1\ 2\ 4) (3) (5\ 9) (6) (7\ 8)$
2 Fixpunkte: $3$ und $6$
\item
\begin{eqnarray*}
s_{n.k} &=& s_{n-1,k-1} + (n-1)s_{n-1,k} \text{ mit} \\
s_{0,0} &=& 1 \\
s_{n,0} &=& 0 \\
s_{n,n} &=& 1 \\\\
s_{9,5} &=& 22449
\end{eqnarray*}
\end{enumerate}
\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{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}
\section*{Aufgabe 2.4}
\begin{itemize}
\item Nur Schritte nach rechts oder oben sind erlaubt
\item Insgesamt $n$ Schritte nach oben und $k$ Schritte nach rechts
\item $\Rightarrow \frac{(n + k)!}{n! k!} = \binom{n + k}{k}$
\end{itemize}
\end{document}

73
school/di-ma/uebung/02/02_2.tex Normal file
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@ -0,0 +1,73 @@
\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 2.2}
\begin{enumerate}[1.]
\item Zyklenzerlegung: $(1\ 2\ 4) (3) (5\ 9) (6) (7\ 8)$
2 Fixpunkte: $3$ und $6$
\item
\begin{tabular}{c|cccccccccc}
$n / k$ & 0&1&2&3&4&5&6&7&8&9 \\\hline
0 & 1 &&&&&&&&& \\
1 & 0 & 1 &&&&&&&& \\
2 & 0 & 1 &1&&&&&&& \\
3 & 0 & 2 & 3 & 1 & & & & & & \\
4 & 0 & 6 & 11 & 6 & 1 & & & & & \\
5 & 0 & 24 & 50 & 35 & 10 & 1 & & & & \\
6 & 0 & 120 & 274 & 225 & 85 & 15 & 1 & & & \\
7 & 0 & 720 & 1764 & 1624 & 735 & 175 & 21 & 1 & & \\
8 & 0 & 5040 & 13068 & 13132 & 6769 & 1960 & 322 & 28 & 1 & \\
9 & 0 & 40320 & 109584 & 118124 & 67284 & \underline{22449} & 4536 & 546 & 36 & 1 \\
\end{tabular}
\begin{eqnarray*}
s_{n.k} &=& s_{n-1,k-1} + (n-1)s_{n-1,k} \text{ mit} \\
s_{0,0} &=& 1 \\
s_{n,0} &=& 0 \\
s_{n,n} &=& 1 \\\\
s_{9,5} &=& 22449
\end{eqnarray*}
\end{enumerate}
\end{document}

84
school/di-ma/uebung/02/02_3.tex Normal file
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@ -0,0 +1,84 @@
\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 &=& 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}

53
school/di-ma/uebung/02/02_4.tex Normal file
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@ -0,0 +1,53 @@
\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 2.4}
\begin{itemize}
\item Nur Schritte nach rechts oder oben sind erlaubt
\item Insgesamt $n$ Schritte nach oben und $k$ Schritte nach rechts
\item $\Rightarrow \frac{(n + k)!}{n! k!} = \binom{n + k}{k}$
\end{itemize}
\end{document}