% This is a TeX file
% It is essentially a programming language for mathematics documents
% You can ignore this first portion of the file, as it is essentially a bunch of "include" statements.
\documentclass[10pt]{article}

\usepackage{pictex, latexsym, graphicx,amsmath,amssymb,amsbsy,amsfonts,amsthm,verbatim}
\usepackage{graphics}
\usepackage{fullpage}
\usepackage{fancyhdr}
\usepackage{hyperref}

\usepackage{algorithm,algorithmic}
\usepackage{multirow}

\setlength{\voffset}{-0.25in}
\setlength{\headsep}{0.25in}
\setlength{\parskip}{1em}
\setlength{\parindent}{0em}

% The following denotes a counter for the problems.
\newcounter{problem}
\newcommand{\problem}{\textbf{\refstepcounter{problem}Problem \theproblem.} }

% These are macro shortcuts! Helpful!
\def\vu{\mathbf{u}}
\def\vx{\mathbf{x}}
\def\vy{\mathbf{y}}
\def\vz{\mathbf{z}}
\def\vb{\mathbf{b}}
\def\vv{\mathbf{v}}
\def\vw{\mathbf{w}}

\renewcommand{\implies}{\rightarrow}
%\newcommand{\lnot}{\neg}
\renewcommand{\lor}{\vee}
\renewcommand{\land}{\wedge}
\newcommand{\xor}{\oplus}
\newcommand{\symmdiff}{\triangle}
\renewcommand{\iff}{\leftrightarrow}
\newcommand{\TRUE}{\mathbf{T}}
\newcommand{\FALSE}{\mathbf{F}}
\newcommand{\universe}{\mathcal{U}}


\def\N{\mathbb{N}}
\def\Q{\mathbb{Q}}
\def\C{\mathbb{C}}
\def\Z{\mathbb{Z}}
\def\R{\mathbb{R}}
\def\cP{\mathcal{P}}


\begin{document}
\pagestyle{fancyplain}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% NOW you can start typing your text that will appear in the PDF!!!

% Note that anything after a '%' is a comment!

% Anything immediately after a '\' is a command. 
% Some denote formatting, some are for special characters.
% The '$' symbol starts and ends "math mode". Use this only for mathematical statements
% In math mode, any sequence of letters is meant to be a list of variables, so use "\sin x" for the sine of x, which will look much different from "sin x"
\chead{COM S 330 --- Homework 10 --- INSERT NAME HERE}

\textit{Type your answers to the following questions and submit a PDF file to Blackboard. 
One page per problem.}


% Pigeonhole
% \clearpage
\problem\ [10pts] Let $n_1,\dots,n_t$ be positive integers. Prove that if $1 + \sum_{i=1}^t n_i$ objects are placed into $t$ boxes, then for some $i$, the $i$th box contains at least $n_i + 1$ objects.

% Pigeonhole
% \clearpage
\problem\ [15pts] Recall the definition of the Ramsey number $r(k,\ell)$.

\textbf{a.} [5pts] Prove that $r(k,2) = k$.

\textbf{b.} [10pts] Use problem 1, part (a), and the fact that $r(3,3) = 6$ to prove that $r(3,4) \leq 10$. 

% Double counting or identities
% \clearpage
\problem\ [10pts] Let $k$ and $n$ be integers with $0 \leq k \leq n$. Give a formula for the coefficient of $x^{2k}$ in the polynomial $(x + \frac{1}{x})^{2n}$. Simplify the formula as much as possible.


% Double counting or identities
% \clearpage
\problem\ [10pts] Give a combinatorial proof that $\sum_{k=0}^n k\binom{n}{k}^2 = n\binom{2n-1}{n-1}$. [Hint: Select a committee of size $n$ from a group of $n$ mathematicians and $n$ computer scientists and select a chair for the committee that is a mathematician.]










\end{document} 