Exercise 5.
How many integer solutions are there to the equation \(x + y + z = 8\) for which
\(x\text{,}\) \(y\text{,}\) and \(z\) are all positive?
\(x\text{,}\) \(y\text{,}\) and \(z\) are all non-negative?
\(x\text{,}\) \(y\text{,}\) and \(z\) are all greater than or equal to \(-3\text{.}\)