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\begin{document}
\title{Math 301 Homework}
\author{Mary Radcliffe}
\date{Due 15 September 2017}
\maketitle

Complete the following problems. Fully justify each response.

\begin{enumerate}
\item Suppose a class contains $2n$ students, and they have projects to work on in pairs.
\begin{enumerate}
\item How many different ways can the professor pair the students up?

\item After the first project, the professor wishes to create new pairings, so that nobody will be working with the same partner a second time. How many ways can the professor create this second set of pairings?
\end{enumerate}

\item Determine the number of ways to create a floral arrangement having $n$ flowers, in which at least 3 flowers are daisies, at least 2 flowers are carnations, and no more than 5 flowers are roses. You may assume every flower is either a daisy, carnation, or rose.


\item Find an ordinary generating function for $(a_n)$ for each of the following recurrence relations:
\begin{itemize}
\item $a_{n+1}=\alpha a_n+\beta$ for $n\geq 0$, $a_0=0$.

\item $a_{n+2}=2a_{n+1}-a_n$ for $n\geq 0$, $a_0=0$, $a_1=1$

\end{itemize}


\end{enumerate}
\end{document}