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

1. [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?

2. [4.4.{8-11}] Solutions to equations.

   1. 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 $x_4\ge 8$.
   2. Count the integral solutions to $x_1+\cdots +x_5=47$ with $5\le x_i\le 30$ for each $i$.
   3. 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?
   4. Find the number of non-negative integer solutions to $x_1+\cdots +x_7\le 47$.
3. 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 $\{2,5,8\}$.