\documentclass[12pt]{article}
\usepackage{amsmath}  % this adds some math 
\usepackage{fullpage}
\usepackage{tikz}            % if you delete this, you will have trouble with the included picture. You can delete the picture.
\usepackage{chessboard}


% If you have a problem, please let me know sooner than later

\newcommand\name{Chuck Norris} %%%%%%%%%%%%%%  WRITE YOUR NAME HERE



\newcounter{excounter}
\setcounter{excounter}{1}
\newcommand\question[2]{\vskip 1em  \noindent\textbf{\arabic{excounter}\addtocounter{excounter}{1}:} \emph{#1} \noindent#2}


% You can also erase this if you do not have package hancyhdr
% Fancy footnote......... 
\usepackage{fancyhdr}  %% If it does not work with your latex installation, you may just delete this...
\pagestyle{fancy}
\usepackage{lastpage}
\rfoot{\name, page \thepage/\pageref{LastPage}}
\cfoot{}
\rhead{}
\lhead{}
\renewcommand{\headrulewidth}{0pt}
\renewcommand{\footrulewidth}{0pt}



%%% Note that you do not have to use this template. You can use your own writeup or make any adjustment to this one you like.
%%% This is provided only for your convenience. 


\begin{document}


{\bf MATH304  \hspace{1cm} HW 6\hfill \name}
\vskip 1em

% left column
\begin{minipage}[t]{0.75\textwidth}
Due \textbf{Oct  8} before class, \textbf{answer without justification will receive 0 points}.
The solution has to be typed (using \LaTeX). 
\end{minipage}
% right box for score column
\begin{minipage}[t]{0.25\textwidth}
\hfill \begin{tabular}{|r|}
\hline
\hskip 3em Score \hskip 3em \\
\hline \\ \\
\hskip 3em /60 \\
\hline
\end{tabular}
\end{minipage}






\question{(P. 155, \#9)}{
Evaluate the sum
\[  \sum_{k=0}^n(-1)^k\binom{n}{k}10^{k}.\]
}

\textbf{Solution:}



\question{(P. 156, \#13)}{
Find one binomial coefficient equal to the following expression:
\[
\binom{n}{k} + 3\binom{n}{k-1} + 3\binom{n}{k-2} + \binom{n}{k-3}.
\]
}

\textbf{Solution:}



\question{(P. 156, \#15)}{
Prove that for every integer $n > 1$,
\[
\binom{n}{1} - 2\binom{n}{2} + 3\binom{n}{3} + \cdots (-1)^{n-1}n\binom{n}{n} = 0.
\]
}

\textbf{Solution:}


\question{(P. 156, \#20)}{
Find integers $a$,$b$, and $c$ such that
\[
m^3 = a\binom{m}{3} + b\binom{m}{2} + c\binom{m}{1}
\]
for all $m$. Then sum the series $1^3 + 2^3 + 3^3 + \cdots + n^3$.
}

\textbf{Solution:}


\question{(P. 157, \#27)}{
Let $n$ and $k$ be positive integers. Give a combinatorial proof of the
identity
\[
n(n+1)2^{n-2} = \sum_{k=1}^{n}k^2\binom{n}{k}
\]
}

\textbf{Solution:}



\question{(P. 157, \#24)}{
Consider a three-dimensional grid whose dimensions are 10 by 15 by 20.
You are at the front lower left corner of the grid and you wish to get to the
back top right corner 45 steps away. How many different routes are there
that take exactly 45 steps?
}

\textbf{Solution:}


 
\end{document}














