\problemdata       % Takes 5 *mandatory* arguments
  {counting-relations}     % latex-friendly label for the prob.
  {counting,functions,relations}  % The topic(s) of the problem content.
  {original}               % Source (if known)
  {S02 q3}                 % Usage (list ones you are aware of).
  {Nick Matsakis, S02}     % Last revision info (author, date).
\begin{problem}[20]~~

For each of the following counting problems, choose the formula which
correctly expresses the answer, and write its number in the box.
\newcommand{\answerbox}[1]{ 
  \hspace{1cm}
  \framebox{\parbox[t][1.5cm][c]{1.5cm}{\centering\insolutions{#1}}}
}

\begin{problemparts}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\parbox{5.5in}{
\problempart[4]The number of functions from a set $A$ of size $n$ to a set $B$
of size $m$.} 
\answerbox{$m^n$}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\parbox{5.5in}{
\problempart[4] The number of \emph{injective}
functions\footnotemark\ from a set $A$ of size $n$ to a set $B$ of
size $m$ (where $m \geq n$).  } 
\answerbox{$\binom{m}{n} \cdot n!$}

\footnotetext{$f(x)$ is \emph{injective} if whenever $a \neq b$, $f(a) \neq f(b)$. In other words, $f(a) = f (b) \implies a = b$}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\parbox{5.5in}{
\problempart[4] The number of undirected graphs with exactly $n$
vertices. Self-loops are not allowed.} 
\answerbox{$2^{\binom{n}{2}}$}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\parbox{5.5in}{
\problempart[4] The number of undirected graphs with exactly $n$ vertices
and exactly $m$ edges. Self-loops are not allowed.
%(where $m \leq \binom {n}{2}$)
} 
\answerbox{$\binom{\binom{n}{2}}{m}$}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\parbox{5.5in}{
\problempart[4]
The number of different\footnotemark\ ice cream cones 
with exactly $m$ scoops when $n$ different flavors are available. 
%($m$ may be larger than $n$).
} 
\answerbox{$\binom{m+n-1}{m}$}
\footnotetext{order of scoops on the cone doesn't matter}

\end{problemparts}

\textbf{Formula Choices}

\begin{enumerate}
\begin{minipage}[b][4cm][b]{0.75in}
	\item $2^{mn}$ 
	\item $m^n$ 
	\item $n^m$ 
	\item $\binom{m+n-1}{m}$ 
	\item $\binom{m+n-1}{n}$   
\end{minipage}\hspace{1in}
\begin{minipage}[b][4cm][b]{0.75in}
	\item $\binom{m}{n}$   
	\item $\binom{n}{m} $ 
	\item $\binom{m}{n} \cdot n!$
	\item $\binom{n}{m} \cdot m!$   
	\item $2^{n^2}$   
\end{minipage}\hspace{1in}
\begin{minipage}[b][4cm][c]{3in}
	\item $2^{\binom{n}{2}}$   
	\item $\binom{\binom{n}{2}}{m}$  
	\item None of the above 
\end{minipage}
\end{enumerate}

\end{problem}

