\documentclass[12pt]{amsart}
% Call packages that allow you to invoke certain mathematical symbols.
\usepackage{amssymb,amsmath,amsthm}
\usepackage[margin=0.9in]{geometry}
\usepackage{cases}
% Set the title, author, and date information.
% Formally begin the document and make the title.
\begin{document}
\noindent \textbf{Math 315 - Fall 2017 \\
Homework 2} \\
Due September 21, 2017 \vspace{-4mm}\\
\noindent \small \textit{It is difficult to find a definition of combinatorics that is both concise and complete, unless we are satisfied with the statement``Combinatorics is what combinatorialists do''}\\\vspace{1mm} \hfill---
W.T. Tutte
\vspace{-5mm}
\normalsize
\noindent \hrulefill
\vspace{3mm}\\
\noindent \textbf{Turn in:}
\vspace{5mm}
\bigskip
\begin{enumerate}
\item (Exercise 23, Chapter 2) Prove that for all positive integers $n$, \[1^3+2^3+ \cdots + n^3 = (1+2+\cdots n)^2\]
\bigskip
\item (Exercise 31, Chapter 2) Let $a_1 = 1$ and let $a_{n+1} = 3 \cdot a_n +4$ for $n\geq 1$. Prove that for all positive integers $n$ the inequality $a_n \leq 3^n$ holds.
\bigskip
\item (Exercise 42, Chapter 3) A host invites $n$ couples to a party. She wants to ask a subset of the $2n$ guests to give a speech, but
she does not want to ask {\em both} members of any couple to give speeches. In how many ways can she proceed? (Hint: your answer shouldn't include a summation sign or a ...)
\bigskip
\item Prove the following identity:
$${n \choose k} - {n-3 \choose k} \;=\; {n-1 \choose k-1} + {n-2 \choose k-1} + {n-3 \choose k-1}$$
{\em Hint:} Use a combinatorial proof.
Let $S$ be a set with three distinguished elements $a$, $b$, and $c$ and count certain $k$-subsets of $S$.
\end{enumerate}
\end{document}