Willy Avoids Fences

Willy lives at the origin and is trying to go to school at Maya Angelou Elementary, which is located at \((6,\,4)\). He needs to get to school by making successive movements one unit to the right or one unit up.

Unfortunately, the evil businessman Rockefeller Tycoon has put up fences between \((1,\,1)\) and \((2,\,1)\) and between \((3,\,4)\) and \((4,\,4)\). Willy can no longer move along those paths.

In how many ways can Willy get to school?

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