King and Queen

Discrete Mathematics Level 3

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

As an explicit example, if the Queen is on \(f2\), the \(f1\), \(f3\), \(f4\), \(f5\), \(f6\), \(f7\), \(f8\), \(g2\), \(h2\), \(e2\), \(d2\), \(c2\), \(b2\), \(e1\), \(g1\), \(g3\), \(h4\), \(e3\), \(d4\), \(c5\), \(b6\), \(a7\) and \(a2\) squares are under attack.

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

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

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