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\begin{document}
\title{Math 301: Homework 1}
\author{Mary Radcliffe}
\date{due 9 Sept 2015}
\maketitle

Complete the following problems. Fully justify each response.

\begin{enumerate}

\item How many words $(a_1, a_2, \dots, a_m)\subset[n]^m$ satisfy $a_1<a_2<\dots<a_m$? 

\item How many integer-valued solutions are there for the following equation, subject to the listed restrictions?
\[ x_1+x_2+x_3+x_4=152\]
\begin{enumerate}
\item  $x_1, x_2, x_3, x_4>0$
\item  $x_1, x_2, x_3>0$ and $x_4\geq 0$
\item  $x_1, x_2, x_3>0$ and $x_4\geq 0$ and $x_2\leq 15$
\end{enumerate}

\item Let $k$ and $n$ be integers such that $0\leq k\leq n-1$. Provide a combinatorial proof of the identity
\[\sum_{j=0}^k{n\choose j}= \sum_{j=0}^k{n-1-j\choose k-j}2^j.\]
(Note: by a ``combinatorial proof," we mean a proof that is based on counting, rather than an algebraic proof.)


\end{enumerate}

\end{document}