\documentclass[11pt]{article}
\usepackage{jeffe,handout,multicol,tikz,kgm}
\usepackage{graphicx}

\newcommand{\kgmcomment}[1]{}
\renewcommand{\gcd}{\mathrm{gcd}}

\begin{document}
\headers{Math378}{Homework 11}{Due Fri. Apr 10, 2026}

%\makeatletter
%\def\@oddfoot{\hfill \tikz \node[draw] {over} ; }
%\makeatother

%\newcommand{\seq}[1]{\left\langle #1\right\rangle}

\newcommand{\RR}{\mathbb{R}}
\newcommand{\ZZ}{\mathbb{Z}}

\makeatletter
\def\@oddfoot{}
\makeatother

\noindent
\textbf{Directions:} Solve the following problems.  All written work must be your own.  See the course syllabus for detailed rules.

\begin{enumerate}

\item {} [4.4.2] I want to buy exactly 10 jars of various herbs and spices, and I am only interested in Cinnamon, Curry, Cumin, Caraway, Coriander, and Chervil.  The supermarket has plenty of each.  How many different combinations are possible?

\item {} [4.4.\{8-11\}]  Solutions to equations.
\begin{enumerate}
	\item Count the integral solutions to $x_1 + x_2 + x_3 + x_4 = 30$ with $x_1\ge 2$, $x_2 \ge 0$, $x_3 \ge -5$, and $x_4\ge 8$.
	\item Count the integral solutions to $x_1 + \cdots + x_5 = 47$ with $5\le x_i \le 30$ for each $i$.
	\item How many non-negative integer solutions are there to $x_1 + \cdots + x_8 = 47$, where exactly three of the variables are equal to zero?  What if we wanted at least three variables equal to zero?
	\item Find the number of non-negative integer solutions to $x_1 + \cdots + x_7 \le 47$.
\end{enumerate}

\item {} How many ways are there to form a subset of $[n]$ of size $k$ with the property that each selected number is at distance at least $3$ from every other selected number?  For example, if $n=8$ and $k=3$ there are $4$ ways: $\{1,4,7\}$, $\{1,4,8\}$, $\{1,5,8\}$, and $\{2,5,8\}$.



\end{enumerate}
\end{document}
