Consider an ant that is walking on a Cartesian grid, starting at (0,0) and ending at (20, 12). The ant always chooses to walk exactly one unit either up or to the right (towards his destination) whenever he arrives at a Lattice point. (A Lattice point is a point with integer coordinates.) Thus, from (0,0) he either walks to (1, 0) or (0, 1). If the ant is not allowed to go to the points (10, 5) and (12, 8), how many different paths can he take on his walk?