A surprise path counting project

This afternoon my son got a pack of rockwool for a science project:

Today I learned what rockwool is. I want to use it for a combinatorics path counting project rather than the school science experiment my son ordered it for! pic.twitter.com/6jG1ahmduj

— Mike Lawler (@mikeandallie) May 8, 2020

I managed to squeeze in a path counting project before he used it!

I started with a question about counting paths that he could solve by listing all of the possibilities:

In the last video my son was getting a little stuck trying to find a way to count the paths, but he did have a few ideas. Here we looked at a different set of paths and were able to find a good counting method:

Finally, with our counting formula in hand, we calculated the number of paths from the bottom left corner to the top right one:

Obviously this wasn’t a planned project, but it was fun to share this neat counting idea!