The problem asks how many ways a sum of 6 can be obtained when rolling three dice.

Each die has 6 faces, so we need to find the number of combinations of rolling three dice that give a total sum of 6.

Step-by-step approach:

1. The minimum value on each die is 1. Therefore, to get a sum of 6, we could explore all combinations where the sum of the three numbers on the dice equals 6.

2. We are looking for solutions to the equation: $x_1 + x_2 + x_3 = 6$ where $x_1, x_2, x_3$ are integers between 1 and 6 (since each die can show values from 1 to 6).

3. We can subtract 1 from each $x_i$ to simplify the problem to finding non-negative integers $y_1, y_2, y_3$ such that: $y_1 + y_2 + y_3 = 3$ where each $y_i \geq 0$. This is a typical stars and bars problem, with the number of solutions given by the binomial coefficient $\binom{3+2}{2} = \binom{5}{2} = 10$.

4. Now, we enumerate the specific dice rolls that result in a sum of 6:

   • (1, 1, 4)
   • (1, 2, 3)
   • (1, 3, 2)
   • (1, 4, 1)
   • (2, 1, 3)
   • (2, 2, 2)
   • (2, 3, 1)
   • (3, 1, 2)
   • (3, 2, 1)
   • (4, 1, 1)

There are 10 ways to obtain a sum of 6 with three dice.

Would you like more details on this, or do you have any further questions?

Related Questions:

1. How many ways can a sum of 7 be obtained with three dice?
2. What is the probability of getting a sum of 6 when rolling three dice?
3. How many ways can a sum of 8 be obtained with three dice?
4. What is the expected sum when rolling three dice?
5. How does the distribution of sums from rolling three dice look?

Tip:

For problems involving dice rolls, breaking the problem down into small cases or using combinatorial methods like stars and bars is often very effective.