Sure. The total number of yellow balls (the ones not in bottom row), is 1+2+3+...+(n-1). But each yellow ball determines exactly two blue balls (the ones in the

Sure. The total number of yellow balls (the ones not in bottom row), is 1+2+3+...+(n-1). But each yellow ball determines exactly two blue balls (the ones in the bottom row): they're the ones you get by travelling left and right from the yellow ball in a straight line until the bottom. You can visually confirm it to be a 1-1 correspondence: different yellow balls will determine different pairs, and each pair is determined by some yellow ball. Therefore the total number of yellow balls is also the number of ways to choose 2 blue balls out of the total number of n blue balls.