"The sum of the first $n$ positive integers is ${n+1 \choose 2}$" is beautiful! for anyone lacking the background to get it, the right hand side is "(n + 1) cho

"The sum of the first $n$ positive integers is ${n+1 \choose 2}$" is beautiful! for anyone lacking the background to get it, the right hand side is "(n + 1) choose 2", the number of ways of selecting 2 elements out of a set of (n + 1). and if you look at the picture, selecting any two balls in the bottom row uniquely identifies a ball in the triangle, and vice versa (selecting a ball in the triangle picks a unique pair of balls in the bottom row). so the sum of all the balls in the first n rows is indeed the number of ways of choosing two balls from the bottom row!