Three distinguishable dice are rolled. In how many ways we can get a total 15?

Answer

To find the number of ways to get a total of 15 when rolling three distinguishable dice, we can follow these steps:

### Step 1: Identify the possible combinations
We need to find combinations of three numbers (each between 1 and 6, inclusive) that add up to 15. The maximum sum we can get with three dice is 18 (6 + 6 + 6), and the minimum is 3 (1 + 1 + 1). Therefore, we need to find combinations that yield a sum of 15.

### Step 2: List the combinations
The combinations that yield a sum of 15 are:
1. (6, 6, 3)
2. (6, 4, 5)
3. (5, 5, 5)

### Step 3: Calculate the arrangements for each combination
1. **For (6, 6, 3)**:
- The number of arrangements is given by the formula for permutations of multiset:
\[ \text{Number of arrangements} = \frac{3!}{2!} = 3 \]
(since there are two 6's and one 3).

2. **For (6, 4, 5)**:
- All numbers are distinct, so the number of arrangements is:
\[ 3! = 6 \]

3. **For (5, 5, 5)**:
- All numbers are the same, so the number of arrangements is:
\[ \frac{3!}{3!} = 1 \]

### Step 4: Sum the arrangements
Now, we sum the number of arrangements from all combinations:
\[ 3 \text{ (from 6, 6, 3)} + 6 \text{ (from 6, 4, 5)} + 1 \text{ (from 5, 5, 5)} = 10 \]

### Conclusion
The total number of ways to get a sum of 15 when rolling three distinguishable dice is **10**.