4 Player Board Game

Discrete Mathematics Level 5

Four players are playing a game involving choosing \(1\times1\) squares on a grid of size \(3 \times 8\). Each player chooses a random square on the grid, then all players reveal their choices and a token is placed in the center of each of these squares. The probability that the tokens form the vertices of a non-degenerate rectangle can be expressed as \(\frac{a}{b}\) where \(a\) and \(b\) are coprime positive integers. What is the value of \(a + b?\)

Details and assumptions

Players are allowed to have selected the same squares. There is no restriction on their choices.
A degenerate rectangle has 0 area.
Squares are rectangles.


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