Carter the Centipede starts at (0, 0, 0) at noon and each minute moves one unit in either the positive x-direction, the positive y direction, or the positive z-direction. Thus, after 1 minute he could be at (1, 0, 0), (0, 1, 0), or (0, 0, 1); after 2 minutes, he could be at (2, 0, 0), (0, 2, 0), (0, 0, 2), (1, 1, 0), (1, 0, 1), or (0, 1, 1). How many different paths could he take to (3, 4, 1) which don't pass through (1, 1, 1)?
