Three fair six-sided dice are thrown. Determine the probability that the sum of the numbers on the three top faces is $6$.
1. **Determine the total number of possible outcomes when rolling three six-sided dice:**
   Each die has 6 faces, so the total number of possible outcomes when rolling three dice is:
   \[
   6 \times 6 \times 6 = 6^3 = 216
   \]

2. **Identify the combinations of dice rolls that sum to 6:**
   We need to find all the combinations of three numbers (each between 1 and 6) that add up to 6. These combinations are:
   - \( (1, 1, 4) \)
   - \( (1, 2, 3) \)
   - \( (2, 2, 2) \)

3. **Count the number of permutations for each combination:**
   - For \( (1, 1, 4) \):
     There are 3 permutations: \( (1, 1, 4), (1, 4, 1), (4, 1, 1) \).
   - For \( (1, 2, 3) \):
     There are 6 permutations: \( (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), (3, 2, 1) \).
   - For \( (2, 2, 2) \):
     There is only 1 permutation: \( (2, 2, 2) \).

4. **Sum the number of favorable outcomes:**
   The total number of favorable outcomes is:
   \[
   3 + 6 + 1 = 10
   \]

5. **Calculate the probability:**
   The probability of rolling three dice and getting a sum of 6 is the number of favorable outcomes divided by the total number of possible outcomes:
   \[
   \frac{10}{216} = \frac{5}{108}
   \]

The final answer is \(\boxed{\frac{5}{108}}\).