Let \(S\) be the set of all \(3\)x\(3\) matrices which have only \(0\)'s and \(1\)'s as entries. (The number of entries that can be \(0\) can be any integer from \(0\) to \(9\) inclusive, as is the case for the number of entries that can be \(1\).)

The probability that a matrix, chosen at random from \(S\), is symmetric is \(\dfrac{a}{b}\), where \(a\) and \(b\) are positive coprime integers. Find \(a + b\).

(This post was inspired by this question.)

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