A Galton board is an upright flat surface with protruding horizontal pins arranged as shown below. At the bottom are a number of bins; if the number of rows is n, the number of bins is n+1. A ball is dropped directly above the top pin, and at each pin bounces left or right with equal probability. We assume that the ball next hits the pin below and immediately left or right of the pin it has struck, and this continues down the board, until the ball falls into a bin at the bottom. If we number the bins from 0 to n, how many paths can a ball travel to end up in bin k?