**0. Keep working on the Term Paper portion of the Final Exam.**To help keep you on track, put a draft of your Take Home Final Exam pdf in Dropbox before Tuesday. It should contain at least five of the six proofs. I won’t grade them for correctness until your final draft but I will give you a homework grade for making progress.**1. Write up and hand in a combinatorial proof for each of the following.**Write your proofs in LaTeX using the math299.sty and put your pdf file in Dropbox as usual. You cannot use any algebra, subsitution, or previous formulas derived in class. Just 100% pure counting!**Theorem A:**$\binom{6}{2}\cdot\binom{4}{3}=\binom{6}{3}\cdot\binom{3}{2}$**Theorem B:**For all combinatorial expressions $n$ and $k$ with $2\leq k\leq n$ $$\binom{2n+2}{k}=\binom{2n}{k}+2\binom{2n}{k-1}+\binom{2n}{k-2}$$**Theorem C:**For all combinatorial expressions $n$ $$\binom{n}{0}+\binom{n}{1}+\binom{n}{2}+\cdots+\binom{n}{n}=2^n$$ [Note: this says that if we add up a row of Pascal’s triangle we always get a power of two. Try it and see for yourself! Of course you can’t use that as a proof, I’m just pointing out that the theorem is cool.]

# Math 299: Introduction to Mathematical Proof

## Daily Activities and Problem Sets:

### Assignment #25 – Thursday, May 11, 2016

#### Course Handouts

- Course Syllabus
- Lecture Notes – check here often for revised lecture notes for our course

#### Proof Software

- Lurch – a math word processor that can check your proofs! I designed and wrote this software with former Scranton math major Nathan Carter specifically with this course in mind.
- 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

#### Mathematical Writing and Typesetting

**Overleaf**- Overleaf – a free website where you can easily produce LaTeX math documents through a web browser
- Homework Template – click this link to start a new assignment (chose ‘Clone this project’ after clicking on this link).
- Sample Document – an example document using the assignment style above.
- 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.

#### Mathematical Typesetting (continued)

**LyX**- LyX – a free math word processor based on LaTeX
**LaTeX for laptops**- MikTeX – install this and TeXnicCenter for LaTeX on your Windows computer
- TeXnicCenter – install this and MikTeX for LaTeX on your Windows computer
**LaTeX for Mac**- MacTeX – Install this for LaTeX on your Mac computer