# Expected number of paths

Discrete Mathematics Level pending

Consider a $$8 \times 8$$ grid of city streets, and let $$S$$ be the points of intersections of the streets and let $$P$$ be the set of paths from the bottom left corner to the top right corner which consist of only walking to the right and up. A point $$s$$ is chosen at random from $$S$$. The expected number of paths in $$P$$ which contain $$s$$ can be expressed as $$a + \frac{b}{c}$$ where $$a,b,c$$ are positive integers, $$b$$ and $$c$$ are coprime, and $$b < c$$. What is the value of $$a + b + c$$?

Details and assumptions

There are 8 rows and columns of city streets, and a total of 64 points of intersection in $$S$$. (This corresponds to 7 city blocks in each direction.)

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