\({19 \choose 2}\) solutions. This problem is equivalent to finding the number of solutions to \(x' + y' + z' = 17\) where \(x'\text{,}\) \(y'\) and \(z'\) are non-negative. (In fact, we really just do a substitution. Let \(x = x'- 3\text{,}\) \(y = y' - 3\) and \(z = z' - 3\)).
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