Homework Set 5

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. Here is an example Overleaf document showcasing this format.
  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).

ChatGPT/AI

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

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.

  1. Suppose your teacher has a set of 500 math problems to distribute amongst the class. If there are 50 people in the class, how many ways can these problems be distributed if:

a. There are no other restrictions

b. Each student must have at least 1 problem.

c. The students are in groups of 5, and the problems are distributed to groups.

d. The problems are distributed evenly amongst the students.

  1. Show that if there are 500 problems and 50 students, at least one student must have 10 or more problems.

  2. Imagine you have a set of dice. There are 4 6-sided dice and 4 8-sided dice.

a. How would you set up generating functions to solve a problem about the number of ways a specific sum could occur when rolling this set of dice? \textbf{Do not attempt to solve this problem, just explain how you would set it up.}

b. If you rolled only the 6-sided dice, What are the maximum and minimum sums?

c. If you rolled only the 6-sided dice, of all possible sums after the dice are rolled, what sum do you think is most common?

d. If you rolled only the 8-sided dice, of all possible sums after the dice are rolled, what sum do you think is most common?

  1. If there is a set of the integers from 1 to 9, any subset of size 6 must contain two numbers that sum to ten.

a. If you have a set of integers from 1 to 11, what can we say instead? What size subset is needed for two numbers to sum to 12?

b. If you have a set of integers from 1 to 13, what can we say now? This time give the size of subset and the sum.

c. What would this problem be for a set of integers from 1 to n?

Suppose you are at a deli. There are 5 types of meat, 4 types of cheese, 3 types of bread, and 7 types of toppings.

a. How many ways are there to make a sandwich with one type of each item?

b. How many ways are there to make a sandwich with two types of toppings, and one type of every other item?

c. How many ways are there to make a sandwich with any combination of toppings? Assume there is one type of each other item on the sandwich.

d. How many ways are there to make a sandwich if you can order double meat or two different kinds of meat? Assume there is one type of each other item on the sandwich.