Aha! Puzzle This
This problem is easiest solved backwards. If we start with a 1x1 square, there are only 2 paths.
From that point, there are 2 other points to go to and both of them only have 1 path to the end, so the number of paths from any given point is just the sum of the paths from the points you would visit next.
Moving back, you can see that there are 3 paths from the point directly to the left.
If we fill in all the values by summing the 2 options from the point one to the right and the point one down, we can build up our matrix until we arrive at the solution.
The total solution comes out to 12,870 paths.