Alice has a new tablet computer. At the login screen, rather than having to input a password, there is a grid of nine points. The login access code is created by dragging your finger from one point to the next in a predefined sequence. You may start at any point and drag your finger to any other point that can be reached directly by a straight line (without going through any intermediate points). You may not use the same point twice.

Assuming that Alice’s access code uses just 4 points (and 3 paths), how many such access codes are possible?

One example is given in the diagram. Also, just to clarify, a direct path between two points is one that does not intersect any other points. So, for example, one cannot go directly from the top-left point to the top-right point; in this case one must pass through the middle-top point. You can do it, but it will count as a 3-points and 2-paths route.

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