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

1. How many circular arrangements of $\{1,\dots ,2n\}$ do not place two even numbers consecutively?
2. [4.2.7] You want to write down a $3\times 4$ matrix whose entries are either 0 or 1. You want the matrix to have 2 or 3 ones (and the rest zeros). How many such matrices are there?
3. [4.2.10] A class of 32 students is asked to split into lab groups. If there are 10 groups of 3 students and one group of 2 students, how many ways can the groups be formed?
4. [4.2.22] In a letter dated Nov 22, 1693, Samuel Pepys asked Isaac Newton a probability question. He posed three scenarios and wanted to know which one had the greatest chance of success. The scenarios were: throwing six dice in hopes of obtaining at least one 6; throwing 12 dice in hopes of obtaining at least two 6’s; and throwing 18 dice in hopes of obtaining at least three 6’s. Calculate the probability of success in each of these cases.

5. Two fours.

   1. In terms of $n$, what is the probability that a number selected from $\{1,\dots ,10^n\}$ contains exactly two digits equal to $4$?
   2. Which value(s) of $n$ maximize the probability of finding exactly two digits equal to $4$?

6. How many ways are there to arrange the letters of MISSISSIPPI:

   1. with no additional restrictions?
   2. [4.3.7] if all four S’s cannot appear consecutively?
   3. if no two S’s can appear consecutively?