Ants

Two ants start at the origin of a coordinate grid. They may only travel right and up. There are walls built on the line $x = 6$ and on the line $y = 6$ , so the ants may not pass through. They stop when they get to point $(6,6)$ . Every time they arrive at a new point (including the origin and the end), they put down a dot. When two dots are put down on the same point, the point is called "dead". If both ants can take any path of their choice to get to the end, how many "combined paths" will result in 4 dead points?