- Write up and hand in a combinatorial proof for each of the following. Write your proofs in LaTeX using the Homework Article style and put your pdf file in Dropbox as usual. You cannot use any algebra, arithmetic, substitution, or previous formulas derived in or out of class. Just 100% pure counting! Bonus points will be awarded for nontrivial asymptote or original figures, but they aren’t required. Put all of the problems in a single document with a title, abstract, and you can cite this web page and the Lecture Notes as a reference.
- 1. Thm 13.16 (Ordered choice) $$\binom{n}{k}\cdot\binom{n-k}{j}=\binom{n}{k+j}\cdot\binom{k+j}{k}$$
- 2. Thm 13.11 (more counting over computing) $$\left(\!\!\!\middle(\genfrac{}{}{0pt}{}{8}{3}\middle)\!\!\!\right)=\left(\!\!\!\middle(\genfrac{}{}{0pt}{}{4}{7}\middle)\!\!\!\right)$$
- 3. Variant of 13.17 (good things come in triples) $$\binom{n+3}{k}=\binom{n}{k}+3\cdot\binom{n}{k-1}+3\cdot\binom{n}{k-2}+\binom{n}{k-3}$$
- 4. Thm 13.30 (Gauss Redux) $$1+2+\cdots+n=\binom{n+1}{2}$$
Math 299: Introduction to Mathematical Proof
Assignment #26 – Thursday, May 14, 2026
Course Handouts
Lecture Notes – a completely revised web verion of our course lecture notes and exercises. This is currently under development and will be revised frequently.
- Previous Lecture Notes – the lecture notes and exercises from previous instances of our course. This is being revised and replaced by the site above, but is still useful.
- Course Syllabus – our course syllabus
Proof Software
- Lurch – is a math word processor that can check your proofs!
- Toy Proofs – a “toy” proof system I developed to introduce students to the concept of formal proofs.
- Circle Dot Game – describing the definitions behind the game.
- Toy Proof Talk Slides – slides from a talk I gave on Toy Proofs at the 2009 Joint Mathematics Meetings
- Scrambling Ice Cream – a word problem based on a variant of the Toy Proof system, Scrambler!.
Mathematical Writing and Typesetting
Overleaf
Homework Template (Article style)
Click this link to start a new assignment (then copy the project in your list of Overleaf projects to make your own copy from my read-only original).
- Sample Document – an example document using the homework assignment style above.
- Homework
Template (Handout style)
Click this link to start a new assignment that is formatted more like a typical LaTeX article (then copy the project in your list of Overleaf projects to make your own copy from my read-only original). - Overleaf – a free website where you can easily produce LaTeX math documents through a web browser
LaTeX
- LaTeX Cheat Sheet – a quick reference that can be printed on one sheet of paper
(posted by winston at stdout.org)
- Detexify – a quick way to look up the name of a math symbol in \LaTeX by drawing it by hand.