Parentheses Numbers (VIDEO!)
Here is a classic puzzle: In how many ways can one arrange parentheses around a sum of N terms so that one is only ever adding two things at a time? [This is a great puzzle for students to do when they are first learning about parentheses!]
It turns out to be tricky, but in this video we come up with a good way of computing these numbers. In part II of this video, we go further and find an actual explicit formula for them!