# How many quadrilaterals can be formed from 12 points on a triangle?

I would like to check if I did this problem correctly so please confirm or correct my method.

There is a triangle ABC with 3 distinct points on side AB, 4 distinct points on side BC, and 5 distinct points on side AC.

What triangle looks like for reference: How many quadrilaterals can be formed from these points?

My method included counting 2 cases: the first being where 2 points were collinear and the other 2 points were on the two remaining sides, the second being where there were 2 points on one side and 2 points on another side, the third side not contributing any points.

This ended up being something like $$3\cdot4\cdot\binom{5}{2} + 3\cdot5\cdot\binom42 + 4\cdot5\cdot\binom32 + \binom32\cdot\binom42 + \binom32\cdot\binom52 + \binom42\cdot\binom52$$

I just want to know if this is a correct method for finding the solution.

Much appreciated!

• You've done just fine; yes, you nailed it. Don't fret so much about the formatting, given it's your first post ? Any way, start trying to learn mathjax (latex, essentially). You might want to start by clicking on "edit" to see how you can format what is displayed now. Actually, no need to accept any answer; you've got the validation you needed. No one really told you anything you already knew! :-) – amWhy Jan 11 '17 at 18:47
• Thank you for your feedback! I will work on my MathJax skills for my future posts :) – Kevin Wang Jan 11 '17 at 21:01

$$3\cdot 4 \binom52 + \cdots$$ looks much nicer.