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\headers{Math375}{Homework 2}{Due: August 27, 2015}

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\textbf{Directions:} Show your work.  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 How many was are there to arrange the letters of 'ECCENTRIC':
\begin{enumerate}
	\item with no additional restrictions?
	\item beginning \emph{and} ending with a C?
	\item beginning \emph{or} ending with a C (or both)?  (Note: CECENTRIC is allowed.)
	\item with all three C's consecutive?
\end{enumerate}

\item Lattice paths from $(0,0)$ to $(8,4)$.
\begin{center}
\begin{tikzpicture}
\draw (0,0) grid (8, 4) ;
\draw (4,2) circle (0.25cm) ;
\end{tikzpicture}
\end{center}
\begin{enumerate}
\item How many lattice paths are there from $(0,0)$ to $(8,4)$ in which each step increases one of the coordinates by $1$?
\item Suppose there is a deadly dragon that lives at the center $(4,2)$.  How many lattice paths from $(0,0)$ to $(8,4)$ avoid the dragon?
\end{enumerate}

\item How many ways are there to partition the integers $\{1,2,3,\ldots,2n\}$ into pairs?  For example, when $n=2$, there are 3 ways: $\{12,34\}, \{13,24\}, \{14,23\}$.  Note that the order of the pairs is not important (so $\{34,12\}$ is the same is $\{12,34\}$ and the order of within pairs is unimportant (so $\{21,34\}$ is the same as $\{12,34\}$).  

Hint: There are several ways to solve this.  Here is one nice way.  Count the number of permutations of $\{1,2\ldots,2n\}$ of length $2n$ in 2 different ways: one directly, and the other using the rule of product.


\item Let $f(n)$ be the number of ordered lists of positive integers that sum to $n$.  For example:
\begin{center}
\begin{tabular}{c|c|c}
$n$ & $f(n)$ & Lists with sum $n$\\
\hline
$1$ & $1$ & $(1)$\\
$2$ & $2$ & $(1,1)$, $(2)$\\
$3$ & $4$ & $(1,1,1)$, $(2,1)$, $(1,2)$, $(3)$
\end{tabular}
\end{center}
It is pretty easy to guess a formula for $f(n)$.  It is more difficult to give an argument that shows your formula for $f(n)$ is correct.  Try to give one by using a graphical representation of the integer $n$ as a line of $n$ dots.  
\end{enumerate}
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