Three dice are rolled simultaneously and the number shown on all the three dice are added, then the total number of possible ways to the a sum of 7is ________?

Options

A. 12
B. 13
C. 15
D. 16

Answer

To find the total number of possible ways to achieve a sum of 7 when rolling three dice, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Possible Combinations**: We need to find all the combinations of three numbers (from 1 to 6, as these are the faces of a die) that add up to 7.

2. **List Combinations**:
- The combinations that sum to 7 are:
- (1, 1, 5)
- (1, 2, 4)
- (1, 3, 3)
- (2, 2, 3)

3. **Count Arrangements for Each Combination**:
- For (1, 1, 5):
- The arrangement can be calculated using the formula for permutations of multiset:
\[ \frac{3!}{2!} = 3 \text{ ways} \]
- For (1, 2, 4):
- All numbers are different, so:
\[ 3! = 6 \text{ ways} \]
- For (1, 3, 3):
- The arrangement can be calculated as:
\[ \frac{3!}{2!} = 3 \text{ ways} \]
- For (2, 2, 3):
- The arrangement can be calculated as:
\[ \frac{3!}{2!} = 3 \text{ ways} \]

4. **Total the Arrangements**:
- Now, we add up all the ways:
\[ 3 \text{ (from 1, 1, 5)} + 6 \text{ (from 1, 2, 4)} + 3 \text{ (from 1, 3, 3)} + 3 \text{ (from 2, 2, 3)} = 15 \text{ ways} \]

### Final Answer:
The total number of possible ways to achieve a sum of 7 when rolling three dice is **15**.

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