344 lines
9.5 KiB
TeX
344 lines
9.5 KiB
TeX
\documentclass[a4paper]{article}
|
|
\usepackage{array}
|
|
\usepackage{adjustbox}
|
|
\usepackage{graphicx}
|
|
\usepackage[left=3cm,right=3cm,top=3cm, bottom=3cm]{geometry}
|
|
\usepackage{amsmath}
|
|
\usepackage{amssymb}
|
|
\usepackage{caption}
|
|
\usepackage[headsepline, footsepline]{scrlayer-scrpage}
|
|
\usepackage{enumerate}
|
|
\usepackage{dsfont}
|
|
\usepackage{float}
|
|
\usepackage{ifthen}
|
|
\usepackage[]{mathtools}
|
|
\usepackage[]{mathbbol}
|
|
\usepackage{multicol}
|
|
\usepackage{enumitem}
|
|
\usepackage[hidelinks]{hyperref}
|
|
\usepackage[]{circuitikz}
|
|
\usepackage{tcolorbox} % Für schöne Boxen
|
|
\usetikzlibrary{circuits.logic.IEC}
|
|
\usepackage{tabularray}
|
|
\usepackage{tikz}
|
|
\usepackage{tkz-graph}
|
|
\usepackage{listings}
|
|
\usepackage{xcolor}
|
|
\usepackage{setspace}
|
|
\usepackage{letltxmacro}
|
|
|
|
|
|
\setlist{itemsep=0.3em, topsep=0.5em, parsep=0pt}
|
|
|
|
\newcommand{\bpf}[1]{%
|
|
\par\vspace{0.8\baselineskip}% Abstand vor der Überschrift
|
|
\noindent\textbf{#1}% Fettgedruckte Überschrift
|
|
\par\vspace{0.3\baselineskip}% Abstand nach der Überschrift
|
|
}
|
|
|
|
|
|
|
|
\renewcommand{\contentsname}{Inhaltsverzeichnis}
|
|
\renewcommand{\figurename}{Grafik}
|
|
\renewcommand{\partname}{Teil}
|
|
\renewcommand{\epsilon}{\varepsilon}
|
|
|
|
\definecolor{darkgrey}{HTML}{232327}
|
|
|
|
% Zählerdefinition mit AUTOMATISCHEM SUBSECTION-RESET
|
|
\newcounter{commoncounter}[subsection]
|
|
\renewcommand{\thecommoncounter}{\thesection.\arabic{commoncounter}}
|
|
|
|
\usepackage{xparse}
|
|
|
|
\newcommand{\answer}[1]{%
|
|
\ifhmode\newline\newline\fi
|
|
\foreach \n in {1,...,#1}{\noindent\rule{\linewidth}{0.4pt}\par\vspace{1em}}
|
|
}
|
|
|
|
|
|
\newcommand{\currentsection}{}
|
|
\newcommand{\currentsubsection}{}
|
|
|
|
\LetLtxMacro{\oldsection}{\section}
|
|
\LetLtxMacro{\oldsubsection}{\subsection}
|
|
|
|
\RenewDocumentCommand{\section}{s o m}{%
|
|
\IfBooleanTF{#1}%
|
|
{ \oldsection*{#3} }% Stern-Variante → unverändert
|
|
{%
|
|
\IfValueTF{#2}%
|
|
{ \oldsection[#2]{#3} }% Optionales TOC-Argument
|
|
{ \oldsection{#3} }%
|
|
\renewcommand{\currentsection}{#3}%
|
|
}%
|
|
}
|
|
|
|
\RenewDocumentCommand{\subsection}{s o m}{%
|
|
\IfBooleanTF{#1}%
|
|
{ \oldsubsection*{#3} }%
|
|
{%
|
|
\IfValueTF{#2}%
|
|
{ \oldsubsection[#2]{#3} }%
|
|
{ \oldsubsection{#3} }%
|
|
\renewcommand{\currentsubsection}{#3}%
|
|
}%
|
|
}
|
|
|
|
|
|
|
|
% Zähler direkt bei Dokumentstart initialisieren
|
|
\AtBeginDocument{\setcounter{commoncounter}{0}}
|
|
|
|
% BOX-DEFINITIONEN mit korrigierter Zählerlogik
|
|
\newtcolorbox{definitionbox}[1][]{
|
|
before title={\refstepcounter{commoncounter}}, % KRITISCH: vor dem Titel!
|
|
title={Definition \thecommoncounter: #1},
|
|
colback=white,
|
|
colframe=white!75!darkgrey,
|
|
fonttitle=\bfseries,
|
|
boxrule=0.6mm,
|
|
coltitle=black,
|
|
rounded corners,
|
|
before skip=10pt,
|
|
after skip=10pt
|
|
}
|
|
|
|
\newtcolorbox{extrataskbox}[1][]{
|
|
before title={\refstepcounter{commoncounter}}, % KRITISCH: vor dem Titel!
|
|
title={Zusatzaufgabe: #1},
|
|
colback=white,
|
|
colframe=white!75!red,
|
|
fonttitle=\bfseries,
|
|
boxrule=0.6mm,
|
|
coltitle=black,
|
|
rounded corners,
|
|
before skip=10pt,
|
|
after skip=10pt
|
|
}
|
|
|
|
\newtcolorbox{merkbox}[1][]{
|
|
title={Merken: #1},
|
|
colback=white,
|
|
colframe=white!75!red,
|
|
fonttitle=\bfseries,
|
|
boxrule=0.6mm,
|
|
coltitle=black,
|
|
rounded corners,
|
|
before skip=10pt,
|
|
after skip=10pt
|
|
}
|
|
|
|
|
|
|
|
\definecolor{codegreen}{rgb}{0,0.6,0}
|
|
\definecolor{codeblue}{rgb}{0,0,0.8}
|
|
\definecolor{codered}{rgb}{0.8,0,0}
|
|
\definecolor{lightgray}{rgb}{0.95,0.95,0.95}
|
|
|
|
\lstdefinestyle{CStyle}{
|
|
language=C,
|
|
basicstyle=\ttfamily\small, % Monospace-Schrift
|
|
keywordstyle=\color{blue}\bfseries, % Schlüsselwörter fett und blau
|
|
stringstyle=\color{red}, % Strings rot
|
|
commentstyle=\color{codegreen}, % Kommentare grün
|
|
backgroundcolor=\color{lightgray}, % Hintergrundfarbe
|
|
numbers=left, % Zeilennummern links
|
|
numbersep=10px, % Abstand zwischen Zeilennummern und Code
|
|
numberstyle=\color{gray}\texttt,
|
|
stepnumber=1, % Zeilennummerierung Schrittweite 1
|
|
frame=single, % Rahmen um den Code
|
|
tabsize=4, % Tabulatorgröße
|
|
breaklines=true, % Zeilenumbruch aktivieren
|
|
captionpos=below,
|
|
showstringspaces=false,
|
|
xleftmargin=15pt, % Linker Rand für den Code (verschiebt alles nach rechts)
|
|
}
|
|
|
|
|
|
\lstdefinestyle{ShellStyle}{
|
|
language=Bash,
|
|
basicstyle=\ttfamily\small, % Monospace-Schrift
|
|
keywordstyle=\color{black}, % Schlüsselwörter fett und blau
|
|
stringstyle=\color{black}, % Strings rot
|
|
commentstyle=\color{codegreen}, % Kommentare grün
|
|
backgroundcolor=\color{lightgray}, % Hintergrundfarbe
|
|
numbers=left, % Zeilennummern links
|
|
numbersep=10px, % Abstand zwischen Zeilennummern und Code
|
|
numberstyle=\color{gray}\texttt,
|
|
stepnumber=1, % Zeilennummerierung Schrittweite 1
|
|
frame=single, % Rahmen um den Code
|
|
tabsize=4, % Tabulatorgröße
|
|
breaklines=true, % Zeilenumbruch aktivieren
|
|
captionpos=below,
|
|
showstringspaces=false,
|
|
xleftmargin=15pt, % Linker Rand für den Code (verschiebt alles nach rechts)
|
|
}
|
|
|
|
|
|
% Umgebung für Listings mit Titel und Zähler
|
|
\newenvironment{codeexample}[1][]{
|
|
\refstepcounter{commoncounter} % Zähler erhöhen
|
|
\lstset{
|
|
style=CSharpStyle,
|
|
caption={Listing \thecommoncounter: #1}, % Titel mit Zähler
|
|
label={listing:\thecommoncounter}
|
|
}
|
|
}{}
|
|
|
|
\makeatletter
|
|
% Internen Zähler von subsubsection auf commoncounter umbiegen
|
|
\let\c@subsubsection\c@commoncounter
|
|
|
|
% Anzeige der Nummer angleichen
|
|
\renewcommand{\thesubsubsection}{\thecommoncounter}
|
|
\makeatother
|
|
|
|
\setlength{\parindent}{0pt}
|
|
|
|
|
|
|
|
\newcommand{\curdate}{09.07.2026}
|
|
\newcommand{\curtitle}{Pascalsches Dreieck}
|
|
|
|
\title{\curtitle}
|
|
\author{Fierke, Janik, Weidlich}
|
|
\pagestyle{scrheadings}
|
|
\date{\curdate}
|
|
\begin{document}
|
|
\ihead{\normalfont\normalsize Mathematik}
|
|
\chead{\normalfont\normalsize \curtitle}
|
|
\ohead{\normalfont\normalsize \curdate}
|
|
\ofoot{\normalfont\normalsize Seite {\pagemark} von \pageref{LastPage}}
|
|
\ifoot{\normalfont\normalsize Fierke, Janik, Weidlich}
|
|
|
|
\section{\curtitle}
|
|
Im Stochastikunterricht ist dir sicherlich schoneinmal die Formel des Binomialkoeffienten begegnet:
|
|
\begin{definitionbox}[Binomialkoeffizient]
|
|
Der Binomialkoeffizient gibt die Anzahl der Möglichkeiten an, aus einer Menge mit $n$ Elementen $k$ verschiedene Elemente auszuwählen:
|
|
\[
|
|
\binom{n}{k}=\frac{n!}{k!(n-k)!}
|
|
\]
|
|
\end{definitionbox}
|
|
|
|
Zum Warmwerden berechnen wir:
|
|
\begin{multicols}{3}
|
|
$\binom{3}{0}=$ \hrulefill\\\\
|
|
$\binom{3}{3}=$ \hrulefill\\\\
|
|
$\binom{7}{3}=$ \hrulefill\\\\
|
|
|
|
$\binom{5}{0}=$ \hrulefill\\\\
|
|
$\binom{5}{5}=$ \hrulefill\\\\
|
|
$\binom{4}{2}=$ \hrulefill\\\\
|
|
|
|
|
|
$\binom{n}{0}=$ \hrulefill { }für $n\in\mathbb{N}$\\\\
|
|
$\binom{n}{n}=$ \hrulefill { }für $n\in\mathbb{N}$\\\\
|
|
|
|
\end{multicols}
|
|
|
|
\begin{enumerate}
|
|
\item Berechne nun nach folgender Regel:
|
|
\begin{center}
|
|
\textit{Die Summe des Steins ist die Summe der darüberliegenden Steine. }
|
|
\end{center}
|
|
\begin{figure}[!ht]
|
|
\centering
|
|
\begin{circuitikz}[x=0.8cm,y=0.8cm]
|
|
|
|
% oberste Reihe
|
|
\draw[dashed,gray!60] (8.875,16.25) rectangle (10.25,14.875);
|
|
\node[text=gray!60] at (9.55,15.5625) {0};
|
|
|
|
\draw (10.25,16.25) rectangle (13,14.875);
|
|
\node[anchor=center] at (11.6,15.5625) {1};
|
|
|
|
\draw[dashed,gray!60] (13,16.25) rectangle (14.375,14.875);
|
|
\node[text=gray!60] at (13.75,15.5625) {0};
|
|
|
|
% zweite Reihe
|
|
\draw[dashed,gray!60] (7.5,14.875) rectangle (8.875,13.5);
|
|
\node[text=gray!60] at (8.175,14.1875) {0};
|
|
|
|
\draw (8.875,14.875) rectangle (11.625,13.5);
|
|
|
|
\draw (11.625,14.875) rectangle (14.375,13.5);
|
|
|
|
\draw[dashed,gray!60] (14.375,14.875) rectangle (15.75,13.5);
|
|
\node[text=gray!60] at (15,14.1875) {0};
|
|
|
|
% dritte Reihe
|
|
\draw[dashed,gray!60] (6.125,13.5) rectangle (7.5,12.125);
|
|
\node[text=gray!60] at (6.8,12.8125) {0};
|
|
|
|
\draw (7.5,13.5) rectangle (10.25,12.125);
|
|
|
|
\draw (10.25,13.5) rectangle (13,12.125);
|
|
|
|
\draw (13,13.5) rectangle (15.75,12.125);
|
|
|
|
\draw[dashed,gray!60] (15.75,13.5) rectangle (17.125,12.125);
|
|
\node[text=gray!60] at (16.375,12.8125) {0};
|
|
|
|
% unterste Reihe
|
|
\draw[dashed,gray!60] (4.75,12.125) rectangle (6.125,10.75);
|
|
\node[text=gray!60] at (5.425,11.4375) {0};
|
|
|
|
\draw (6.125,12.125) rectangle (8.875,10.75);
|
|
|
|
\draw (8.875,12.125) rectangle (11.625,10.75);
|
|
|
|
\draw (11.625,12.125) rectangle (14.375,10.75);
|
|
|
|
\draw (14.375,12.125) rectangle (17.125,10.75);
|
|
|
|
\draw[dashed,gray!60] (17.125,12.125) rectangle (18.5,10.75);
|
|
\node[text=gray!60] at (17.875,11.4375) {0};
|
|
|
|
\end{circuitikz}
|
|
\end{figure}
|
|
|
|
\item Berechne die Steine der Pyramide:
|
|
\begin{figure}[!ht]
|
|
\centering
|
|
\begin{circuitikz}[x=0.8cm,y=0.8cm]
|
|
|
|
\draw (10.25,16.25) rectangle (13,14.875);
|
|
\node at (10.875,15.5625) {$\binom{0}{0}=$};
|
|
|
|
\draw (8.875,14.875) rectangle (11.625,13.5);
|
|
\node at (9.5,14.1875) {$\binom{1}{0}=$};
|
|
|
|
\draw (11.625,14.875) rectangle (14.375,13.5);
|
|
\node at (12.25,14.1875) {$\binom{1}{1}=$};
|
|
|
|
\draw (10.25,13.5) rectangle (13,12.125);
|
|
\node at (10.875,12.8125) {$\binom{2}{0}=$};
|
|
|
|
\draw (7.5,13.5) rectangle (10.25,12.125);
|
|
\node at (8.125,12.8125) {$\binom{2}{1}=$};
|
|
|
|
\draw (13,13.5) rectangle (15.75,12.125);
|
|
\node at (13.625,12.8125) {$\binom{2}{2}=$};
|
|
|
|
\draw (11.625,12.125) rectangle (14.375,10.75);
|
|
\node at (12.25,11.4375) {$\binom{3}{0}=$};
|
|
|
|
\draw (8.875,12.125) rectangle (11.625,10.75);
|
|
\node at (9.5,11.4375) {$\binom{3}{1}=$};
|
|
|
|
\draw (6.125,12.125) rectangle (8.875,10.75);
|
|
\node at (6.75,11.4375) {$\binom{3}{2}=$};
|
|
|
|
\draw (14.375,12.125) rectangle (17.125,10.75);
|
|
\node at (15,11.4375) {$\binom{3}{3}=$};
|
|
\end{circuitikz}
|
|
\end{figure}
|
|
\end{enumerate}
|
|
\begin{extrataskbox}
|
|
Entwirf einen Computeralgorithmus, der den Binomialkoeffizienten berechnet, ohne dass dieser die Fakultät benutzt!
|
|
\end{extrataskbox}
|
|
|
|
\label{LastPage}
|
|
\end{document}
|