  \documentclass[11pt]{article}
\usepackage{jeffe,handout,multicol,tikz}

\newcommand{\kgmcomment}[1]{}

\begin{document}
\headers{Math375}{Homework 1}{Due: August 20, 2015}

%\makeatletter
%\def\@oddfoot{\hfill \tikz \node[draw] {over} ; }
%\makeatother

\makeatletter
\def\@oddfoot{}
\makeatother

%\noindent
%\hfill Name: \underline{\hspace{3.5in}} 

%\begin{flushright}
%Name: \underline{\hspace{3.5in}} \\
%\end{flushright}
%\hrule

\def\columnseprule{0.4pt}

\noindent
\textbf{Directions:} You may work to solve these problems in groups, but all written work must be your own.  See ``Guidelines and advice'' on the course webpage for more information.

\begin{enumerate}

\item A sandwich shop offers 5 choices of bread, 3 choices of seasonings, 6 choices of meat.  Also, the shop offers 8 toppings: lettuce, spinach, tomato, cheese, avocado, onion, cucumber, and bell pepper.
\begin{enumerate}
\item How many different sandwiches can be ordered at the shop?  If you ate one sandwich a day and never repeated orders, how long would it take you to try them all?
\item The shop owner decides that having both lettuce and spinach at the same time makes the sandwich too green, and makes a rule that a sandwich cannot have both of these toppings.  How many different sandwiches can be ordered now?
\end{enumerate}

\item How many 5-digit ATM pin numbers have no repeated digits within  distance 2 of one another?  For example, 56759 counts because the pairs of 5's are at distance 3 from one another.  Similarly, 56756 counts, but 56765 does not count because the 6's are too close to each other.

\item A library has $n$ different books and two shelves.  How many ways can they arrange the books on the shelves so that no shelf is empty?  

For example, with 3 books (called 1, 2, and 3), there are 6 ways to do this with one book on the first shelf and two on the second: $1|23$, $1|32$, $2|13$, $2|31$, $3|12$, $3|21$.  Since there are also 6 ways to do this with \emph{two} books on the first shelf and \emph{one} on the second, the answer when $n=3$ is 12.
\end{enumerate}
\end{document}
