King and Rook

Discrete Mathematics Level 3

Dimitri places a Black King and a White Rook on an empty chessboard. If the probability that Dimitri places the King and the Rook on the chessboard such that the King is NOT in check (that is, the Rook is not attacking the Black King), can be expressed as \(\frac{m}{n}\), in which \(m\) and \(n\) are coprime positive integers, find \(n-m\).

Details and Assumptions: e.g. If the Rook is on \(f2\), the \(f1\), \(f3\), \(f4\), \(f5\), \(f6\), \(f7\), \(f8\), \(g2\), \(h2\), \(e2\), \(d2\), \(c2\), \(b2\) and \(a2\) squares are under attack.

The King and the Rook cannot be placed in the same square.

This is the fourth problem of the set Look after the King!

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