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How many integer solutions are there to the equation \(x + y + z = 8\) for which

1. \(x\text{,}\) \(y\text{,}\) and \(z\) are all positive?

2. \(x\text{,}\) \(y\text{,}\) and \(z\) are all non-negative?

3. \(x\text{,}\) \(y\text{,}\) and \(z\) are all greater than or equal to \(-3\text{.}\)