Here are the review problems from Thursday’s class.

I’ve written up brief summaries of the solutions (note: these are a rush job and should not be taken as the level of exposition that would be appropriate on homework or a test). The solutions are in white text below; select with the mouse to view. This is so that you can read one solution at a time without having the rest spoiled for you. The solutions to #18, 26, and 27 are absent for now due to constraints on my time.

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Graded: 7.4 #10, 29

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The second midterm will be this Friday, August 1, from 2:00 to 4:00 in 4 Evans. We’ll start as close to 2:00 as possible. The midterm is designed to take about 1 hour.

As before, the only items you may bring are pens/pencils, erasers, and pencil sharpeners. Paper will be provided.

This midterm will cover sections 4.4, 5.1-5.3, 6.1-6.5, 7.1-7.4, and 8.1. You may need to use skills and knowledge developed in the earlier sections as well.

We’ll use Thursday’s class for review, with particular emphasis on counting, combinatorial proofs, and strong induction (as some of you have requested). You are welcome to make additional requests for coverage! Also, I have office hours after class on Wednesday and before class on Friday, as usual.

Sample midterms:

• Prof. Williams’ Spring 2014 sample midterm (solutions). Problem #1(b) is out of bounds.
• Prof. Sturmfels’ Spring 2012 test (solutions).
• Prof. Haiman’s Fall 2008 test (solutions).

Good luck studying!

Graded: Supplemental #1, 7.2 #10

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Graded: 6.3 #24, 6.4 #22

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Part of today’s lesson was a departure from the textbook. Here are my lecture notes. (Note that these are just the “cues” I use in class, which are designed primarily for my own use. They don’t go into full detail and aren’t a good substitute for being in class, but they’re better than nothing.)

These problems are part of the homework due on Monday, July 28.

1. Determine the largest integer $n$ that can’t be written as $5s+7t$, where $s$ and $t$ are nonnegative integers. Prove your answer, i.e., show that your $n$ can’t be written in this form and show that all larger integers can be written in this form.
2. How many paths are there from point A to point B in the grid below, if we can only travel along the lines going north (up) and east (right)? Be sure to notice the “gap” in the grid.
3. Determine how many ways 8 indistinguishable objects can be distributed to 4 indistinguishable boxes. List the ways.
4. Determine how many ways 8 distinguishable objects can be distributed to 4 indistinguishable boxes. Hint: Solve problem 3 first.

Graded: 5.2 #12, 5.3 #12

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Graded: 5.1 #10, #22

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Solutions to Midterm 1 are available. The median score was 77.5.

Please review the grading policy on the syllabus. I have added information about what numerical grades translate to A’s, B’s, C’s, and D’s.