Week 5: Self-Guided Problems

When you come to class, be ready to discuss these questions, and bring any additional questions you still have after this exercise!

Submitting Your Work

Your work must be submitted to Anchor for degree credit and to Gradescope for grading.

For any work completed outside of GitHub or Gradescope:

  1. Typeset your work in LaTeX.
  • You should use \begin{enumerate} to create a numbered list, and your solutions to each problem should match with the corresponding number on the assignment.
  • For example, if an assignment has 10 numbered problems, your enumerate should have 10 \item commands, with your solution(s) to problem 1 under the first \item, problem 2 under the second and so on.
  • If you skip a problem, just leave the \item blank, otherwise the numbers of your solutions will not match the assignment document.
  1. Compile into a PDF. This can be done easily through an Overleaf account, otherwise you will need to install LaTeX.
  2. Submit the pdf to Gradescope via the appropriate submission link for the course.
  3. Upload the pdf to Anchor using the form below.

Note: Anchor submissions can occur at any time during the term, but it is critical that you upload all of your work to Anchor before the last day of the term. Gradescope submissions must be submitted before the deadline (or the late deadline, if applicable).

It is required that all assignments are submitted as a PDF generated from a LaTeX document.

Assignments submitted in any other form will earn zero credit.

ChatGPT/AI

Use of ChatGPT/AI is forbidden for all assignments in this course.

Self-Guided Problems cannot be submitted late for any reason

This is due to the fact we discuss them in the class. Be sure to stay on top of these Self-Guided problems, and remember it is better to turn in an incomplete set than an empty one.

Problems

  1. Suppose we want to count:

  • The number of ways to rearrange the letters KIBO.

  • The number of ways to rearrange the letters KIBO where K is not the first letter.

  • The number of ways to rearrange the letters KIBO where none of the letters are in the position they are currently in.

Which of these problems should have the largest answer? Which is the smallest?

  1. Solve all of the problems from problem 1. What technique did you use to solve them?

  2. Consider the act of rolling 2 6-sided dice labeled 1,2,3,4,5, and 6 on their faces. Create a chart of the possible sums of these dice when rolled. What do you notice about this chart? How would you have counted the number of ways to roll a specific sum (say, 7)?

  3. Suppose there are 50 people in our class. Using pigeonhole principle, find the following:

a. What is the minimum number of people who were born on the same day of the week?

b. What is the minimum number of people that are born in the same month?

c. What is the minimum number of people that have the same eye color?