## A scary approach and a not-so-scary approach to a challenging math problem

Today was the last day of math projects for at least a week because of some vacation and work travel. Rather than jumping ahead in our book I thought it would be neat to show them an interesting identity in Pascal’s triangle.

Prove that for any integer n greater than 0:

${n \choose 0}^2 + {n \choose 1}^2 + \ldots {n \choose n}^2 = {2n \choose n}$

Instead of taking the scary approach, we started today’s problem by talking about counting paths in a square grid. We explored this type of question for the first time in yesterday’s project:

What learning math sometimes looks like: Counting paths in a grid

Here’s what the boys thought about counting these paths today followed by their approach to counting a few subsets of the total paths:

In the last video we started counting paths that passed through specific points in the diagonal of the grid. Here we finish off that calculation to find an interesting, though fairly complicated-looking expression:

For the last part of the project we take a look at what this identity means in terms of Pascal’s triangle. The fairly easy to see relationship here is a nice surprise! Once the boys see the surprise, they are able to find other cases of this identity in Pascal’s triangle

So, a fun project before we take a break for a week. The connections with counting and Pascal’s triangle really are amazing 🙂