Final Portfolio
MAT202, Fall 2023
December 5, 2023
(Introduction) Instructions: Write a short introduction to your portfolio that reflects on your writing process this semester. Say which four questions you picked to revise, and explain your choices—why did you pick these four? What was it about those submissions that made you feel like you could still make a difference in them, and what kinds of improvements did you make for each one?
As you are going back and reading your assignments from earlier in the semester, how do you feel about the progress you’ve made?
(A1) Count the number of arrangements of the word OVERLEAF where
. The string LOVE appears somewhere in the word;
. the two E’s are not beside each other; and
. the letter R appears somewhere to the left of the letter F (but not necessarily beside each other).
(A2) From a standard deck of cards, determine the number of seven-card hands that contain at least two pairs and at least three different suits.
For example, the hand 
is a valid hand.
(A three-of-a-kind or four-of-a-kind does not count as a pair.)
(A3) Write a combinatorial proof of the following identity:
(A4) Let R1 and R2 be two equivalence relations on a set S. Determine and argue whether or not each of these relations are necessarily equivalence relations on S:
. x ∼ y if and only if (x,y) ∈ R1 and (x,y) ∈ R2 .
. x ⋆ y if and only if (x,y) ∈ R1 or (x,y) ∈ R2 (or both).
(A5) Solve the congruence 165x = 375 (mod 410), and express your final answer in terms of congruence classes in Z410. Show and explain all steps in your calculation, including finding an inverse using theEuclidean algorithm.
(A6) For each sequence below, give a simple graph with that degree sequence, or argue that one doesn’t exist.
(a) 6, 4, 4, 4, 4, 4, 4
(b) 7, 5, 4, 2, 2, 2, 1, 1
(Conclusion) Instructions: Wrap up your final portfolio by writing some concluding thoughts. You may also use some of these guide questions to aid your reflection:
. Over the semester,what have been the most significant changes to the way you write mathematics, if any?
. Were you satisfied with your performance in the course? With respect to writing? or the material?
. Have your beliefs on the importance of communication in mathematics changed? How did they change?
. How valuable was the TA feedback in your process of preparing this portfolio?
. How was your experience learning LATEX?
. ...and anything else you’d like to mention!