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\headers{Math375}{Homework 4}{Due: Sept 10, 2015}

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\textbf{Directions:} You may work to solve these problems in groups, but all written work must be your own.  \textbf{Show your work}; See ``Guidelines and advice'' on the course webpage for more information.

\begin{enumerate}

\item How many 5-digit ATM pin numbers:
\begin{enumerate}
	\item have distinct digits that increase from left to right?  (So 02379 counts, but 02279 and 20458 do not.)
	\item have digits that are non-decreasing from left to right?  (So 02379 and 02279 count, but 20458 does not.)
\end{enumerate}

\item \emph{Solutions to equations.}  Count the number of non-negative integral solutions to the following equations.  

\begin{enumerate}
\item $x_1 + x_2 + \cdots + x_6 = 50$
\item $x_1 + x_2 + \cdots + x_6 = 50$ where each $x_i$ is at least $4$
\item $x_1 + x_2 + \cdots + x_6 = 50$ where $x_1 \le 20$
\item $x_1 + x_2 + \cdots + x_6 = 50$ where $1 \le x_i \le 30$ for all $i$.
\end{enumerate}

\item \emph{Lattice paths with diagonal steps.}  A \emph{diagonal step} in a lattice path moves $1$ unit in the $x$-direction and $1$-unit in the $y$ direction.
\begin{center}
\begin{tikzpicture}
\draw (0,0) grid (5,5) ;
\draw (-3ex, 0) node {$(0,0)$} ;
\draw (5, 5) +(3ex, 0) node {$(5,5)$} ;
\draw[line width=3pt] (0,0) -- (1,0) -- (2,0) -- (2,1) -- (3,2) -- (3,3) -- (3,4) -- (3,5) -- (4, 5) -- (5,5) ;
\end{tikzpicture}
\end{center}
\begin{enumerate}
\item For each $k$ with $0\le k\le 5$, determine the number of lattice paths with diagonal steps from $(0,0)$ to $(5,5)$ that have exactly $k$ diagonal steps.  (A lattice path from $(0,0)$ to $(5,5)$ with $1$ diagonal step is displayed above.)
\item Add your results from part (a) to determine the total number of lattice paths from $(0,0)$ to $(5,5)$ with diagonal steps.
\item Using $\Sigma$ notation, give a summation formula for the number of lattice paths with diagonal steps from $(0,0)$ to $(n,n)$.
\end{enumerate}
\end{enumerate}
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