Given positive integers $a, b,$ and $c$ with $a + b + c = 20$.
Determine the number of possible integer values for $\frac{a + b}{c}.$
1. Given the equation \(a + b + c = 20\), we can let \(d = a + b\). Therefore, we have \(d + c = 20\).
2. We need to determine the number of possible integer values for \(\frac{a + b}{c} = \frac{d}{c}\).
3. Since \(d + c = 20\), we can express \(d\) as \(d = 20 - c\).
4. We need to find the integer values of \(\frac{d}{c} = \frac{20 - c}{c}\).
5. Simplify the expression: \(\frac{20 - c}{c} = \frac{20}{c} - 1\).
6. For \(\frac{20}{c} - 1\) to be an integer, \(\frac{20}{c}\) must be an integer. Therefore, \(c\) must be a divisor of 20.
7. The positive divisors of 20 are \(1, 2, 4, 5, 10, 20\).

Now, we check each divisor:
- If \(c = 1\), then \(d = 20 - 1 = 19\) and \(\frac{d}{c} = \frac{19}{1} = 19\).
- If \(c = 2\), then \(d = 20 - 2 = 18\) and \(\frac{d}{c} = \frac{18}{2} = 9\).
- If \(c = 4\), then \(d = 20 - 4 = 16\) and \(\frac{d}{c} = \frac{16}{4} = 4\).
- If \(c = 5\), then \(d = 20 - 5 = 15\) and \(\frac{d}{c} = \frac{15}{5} = 3\).
- If \(c = 10\), then \(d = 20 - 10 = 10\) and \(\frac{d}{c} = \frac{10}{10} = 1\).
- If \(c = 20\), then \(d = 20 - 20 = 0\) and \(\frac{d}{c} = \frac{0}{20} = 0\).

Thus, the possible integer values for \(\frac{a + b}{c}\) are \(19, 9, 4, 3, 1, 0\).

The final answer is \(\boxed{6}\).