If we fill in the entire grid including the top left corner, the bottom right corner and the center, a path from to would take steps upward and steps to the right. How many different paths from to is given by how many ways can we choose steps to the right out of steps?
We need to subtract from the number of paths that go through the points in the top left corner, the points of the right bottom corner, and the central point of the grid. With a little patience and a pencil and paper, we find out that paths go through the top left corner; paths through the right bottom corner; paths through the central point from the left hand point ; and paths through the central point from the bottom point .
Number of different paths from to equals
We collapse the empty central square and thus obtain a grid as follows
A path from to would take steps of which are upward and to the right. How many ways can we choose steps to the right out of steps is given by
We subtract the paths going through and and obtain