\documentclass[11pt] {letter}
\usepackage{graphicx,color}
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\setlength{\textheight}{9in}
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\newtheorem{theorem}{Theorem}        % some macros
\newtheorem{problem}{Problem}        % some macros
\newcommand{\Frac}[2]{\displaystyle \frac{#1}{#2}}
\newcommand{\binom}[2]{\displaystyle \left(\begin{array}{c} 
                                      #1 \\#2 \end{array}\right)}

\begin{document}
\date{~} 
 

\name{Carl Eberhart}
\address{ 
\begin{minipage}{520 pt}
\parbox[t]{250pt}{\includegraphics[height=140pt,width=250pt]{hrswc6.eps}}
\hfill
\parbox[t]{260pt}{\vspace{-100pt} {\Large\sc University of Kentucky\\
Mathematics Dept.\\ Lexington, Kentucky 40503}\\\vfill February 23, 1998} 
\end{minipage} 
}


 
\signature{Carl Eberhart}
\begin{letter}{Bill Smith\\34 Watertown\\Portsmouth, England}
\opening{Dear Bill:}
I wanted to send you some sample equations and graphics to show how nicely
the \LaTeX program formats things.  For example, the letterhead here is
an included graphic of the Horse head Nebula.  If I had a color printer you 
would be getting a color version of the graphic.

Also, here is a statement of the
binomial thereom.

\begin{theorem}[The Binomial Theorem]
Let $n$ and $i$ be integers with $n > 2$ and $1<i<n$. Then $\binom{n}{i}$,
the coefficient of $x^{n-i}y^i$ in the expansion of $(x+y)^n$,  is the sum of the
coefficients of $x^{n-i}y^{i-1}$ and $x^{n-i-1}y^{i}$ in the expansion of 
$(x+y)^{n-1}$.  That is,
\[  \binom{n}{i} = \binom{n-1}{i-1} + \binom{n-1}{i} \]
\end{theorem}


{\bf Proof:}  We induct on $n$.
The statement is true if $n=3$ and $i=2$.  Suppose it is true for
$n-1$ and all $i$ between $1$ and $n-1$. 
We can write $(x+y)^{n } = (x+y)(x+y)^{n-1} = x(x+y)^{n-1}+y(x+y)^{n-1}$.
By induction,  (I do not have room for the proof in the margin)


\closing{Sincerely,}
\end{letter}
\end{document}




