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+\usepackage{titling} +\usepackage[utf8]{inputenc} +\usepackage[english]{babel} +\usepackage{amsthm} +\usepackage{amsmath} +\usepackage{amssymb} +\usepackage{graphicx} +\usepackage{changepage} +\usepackage{enumerate} + +\newtheorem{theorem}{} +\newtheorem{definition}{Definition} +\newtheorem{exercise}{Exercise} +\newtheorem{example}{Example} +\newtheorem{problem}{Problem} +\newtheorem*{corollary}{Corollary} + +\makeatletter +\newenvironment{chapquote}[2][2em] + {\setlength{\@tempdima}{#1}% + \def\chapquote@author{#2}% + \parshape 1 \@tempdima \dimexpr\textwidth-2\@tempdima\relax% + \itshape} + {\par\normalfont\hfill--\ \chapquote@author\hspace*{\@tempdima}\par\bigskip} +\makeatother + +\title{\textbf{Counting in two ways}} +\date{Week 7} +\author{Miroslav Stankovic \\ Marko Puza} +\begin{document} +\maketitle + + +\section{Theory} +Being able to employ a combinatorial point of view in seemingly non-related problems may often prove very useful - even provide a proof. In the following, we will take a look at a number of identities that can be proved by counting in two different ways. The simple example of usefulness of this technique can be the Handshaking lemma. + +\begin{example}[Handshaking lemma] +For any undirected graph, we have $\sum_{v \in V} deg(v) = 2|E|$. \\ (where we are summing over all vertices $v$, $deg(v)$ is the number of edges connected to vertex $v$ and $|E|$ is the number of edges in the graph) +\end{example} +\begin{proof} +We will prove this lemma by counting in two ways. The number of edge-vertex connections is $\sum_{v \in V}deg(v)$ when we take a look at the connections of each vertex. At the same time, if we take a look at the connections of each edge, the number of edge-vertex connections is $2|E|$. +\end{proof} + +\noindent Let us now define the following, which are most probably very familiar to you, combinatorially. + +\begin{definition}[Binomial numbers] ${n}\choose{k}$ is the number of ways how to choose $k$ elements out of $n$. +\end{definition} + +\begin{definition}[Fibonacci numbers] $F_n$ is the number of ways to fill a table of size $(n - 1) \times 1$ by tiles of size $1 \times 1$ and $2 \times 1$. +\end{definition} + +\begin{exercise} +Convince yourself that the above definitions agree with the usual definitions of a binomial and Fibonacci numbers. +\end{exercise} + +\section{Problems} + +\begin{problem} +See that the number of subsets of a set with $n$ elements is $2^n$. +\end{problem} + +\begin{problem} +See that ${n \choose k} = {n \choose n - k}$. +\end{problem} + +\begin{problem} +See that ${n+1 \choose k+1} = {n \choose k} + {n \choose k+1}$. +\end{problem} + +\begin{problem} +See that $n{n-1 \choose k-1}= k{n \choose k}$. +\end{problem} + +\begin{problem} +See that $\sum_{i=1}^{n} i = {n+1 \choose 2}$. +\end{problem} + +\begin{problem} +See that ${n \choose r} = {n - 2 \choose r - 2} + 2{n - 2 \choose r - 1} + {n - 2 \choose r}$. +\end{problem} + +\begin{problem}[Binomial theorem] +See that $(x+y)^n = \sum_{k=0}^n {n \choose k}x^{n-k}y^k$. +\end{problem} + + +\begin{problem} +See that $\sum_{i=1}^n i^2 = 2 {n+1 \choose 3} + {n+1\choose 2}$. Can you derive a formula for $\sum_{i=1}^n i^3$? +\end{problem} + +\begin{problem} +See that $\sum_{k=0}^n {2k \choose k}{2(n-k)\choose n-k} = 2^{2n}$. +\end{problem} + +\begin{problem} +See that ${n \choose r}{r \choose k}= {n \choose k}{n-k \choose r-k}$. +\end{problem} + +\begin{problem} +See that ${2n \choose 2} = 2{n \choose 2} + n^2$. +\end{problem} + +\begin{problem} +See that $F_n + F_{n+1} = F_{n+2}$. +\end{problem} + +\begin{problem} +See that the number of ways to fill a table $(n - 1)\times 2$ by tiles of size $2 \times 1$ is $F_n$. +\end{problem} + +\begin{problem} +See that $F_{a+b+1} = F_{a+1}F_{b+1} + F_{a}F_{b}$. +\end{problem} + +\begin{problem} +See that $F_{0} + F_{1} + \dots + F_{n} = F_{n+2} - 1$. +\end{problem} + +\begin{problem} +See that $F_{1} + F_{3} + \dots + F_{2n-1} = F_{2n} - 1$. +\end{problem} + +\begin{problem} +See that $nF_{0} + (n - 1)F_{1} + \dots + F_{n-1} = F_{n+3} - (n+3)$. +\end{problem} + +\begin{problem} +See that $F_{2n + 1} = F_n^2 + F_{n+1}^2$. +\end{problem} + +\begin{problem}[Number of divisors] +See that for number $n$ with prime factorization $n = p_1^{a_1}p_2^{a_2}\dots p_k^{a_k}$ the number of its divisors is $(a_1 + 1)(a_2 + 1) \cdots (a_k + 1)$. +\end{problem} + +\end{document}