Find the difference between the greatest and least values of $lcm (a,b,c)$, where $a$, $b$, and $c$ are distinct positive integers between $1$ and $10$, inclusive.
1. **Identify the range of values for \(a\), \(b\), and \(c\)**:
   - \(a\), \(b\), and \(c\) are distinct positive integers between 1 and 10, inclusive. Therefore, \(a, b, c \in \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\).

2. **Find the minimum value of \(\text{lcm}(a, b, c)\)**:
   - The least common multiple (LCM) of a set of numbers is the smallest positive integer that is divisible by each of the numbers in the set.
   - To minimize \(\text{lcm}(a, b, c)\), we should choose numbers that share common factors. The smallest distinct numbers are 1, 2, and 3, but their LCM is 6. We need to find a smaller LCM.
   - Consider \(\{1, 2, 4\}\):
     \[
     \text{lcm}(1, 2, 4) = \text{lcm}(\text{lcm}(1, 2), 4) = \text{lcm}(2, 4) = 4
     \]
   - Therefore, the minimum value of \(\text{lcm}(a, b, c)\) is 4.

3. **Find the maximum value of \(\text{lcm}(a, b, c)\)**:
   - To maximize \(\text{lcm}(a, b, c)\), we should choose numbers that are relatively prime to each other, as this will result in a larger LCM.
   - Consider \(\{7, 8, 9\}\):
     \[
     \text{lcm}(7, 8, 9) = \text{lcm}(\text{lcm}(7, 8), 9)
     \]
     - First, find \(\text{lcm}(7, 8)\):
       \[
       \text{lcm}(7, 8) = 7 \times 8 = 56 \quad \text{(since 7 and 8 are relatively prime)}
       \]
     - Next, find \(\text{lcm}(56, 9)\):
       \[
       \text{lcm}(56, 9) = \frac{56 \times 9}{\gcd(56, 9)}
       \]
       - Since 56 and 9 are relatively prime, \(\gcd(56, 9) = 1\):
       \[
       \text{lcm}(56, 9) = 56 \times 9 = 504
       \]
   - Therefore, the maximum value of \(\text{lcm}(a, b, c)\) is 504.

4. **Calculate the difference between the maximum and minimum values**:
   \[
   504 - 4 = 500
   \]

The final answer is \(\boxed{500}\)