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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