Three dice are thrown at the same time. What is the probability that the sum of three numbers that turn up is 15?

Options

A. `(11)/(216)`
B. `(5)/(108)`
C. `(9)/(216)`
D. `(1)/(18)`

Answer

To find the probability that the sum of the numbers on three thrown dice is 15, we can follow these steps:

### Step 1: Determine the total number of outcomes when throwing three dice.
When one die is thrown, there are 6 possible outcomes (1 through 6). For three dice, the total number of outcomes is calculated as:
\[ \text{Total Outcomes} = 6 \times 6 \times 6 = 6^3 = 216 \]

### Step 2: Identify the favorable outcomes that yield a sum of 15.
Next, we need to find all the combinations of numbers on the three dice that add up to 15. We can systematically list the combinations:

1. (6, 6, 3)
2. (6, 5, 4)
3. (5, 5, 5)
4. (6, 4, 5)
5. (4, 6, 5)
6. (5, 6, 4)
7. (3, 6, 6)
8. (5, 4, 6)
9. (4, 5, 6)
10. (3, 5, 7) - Invalid as 7 is not possible on a die.

After considering all combinations, we find the following valid combinations that sum to 15:
- (6, 6, 3)
- (6, 5, 4)
- (5, 6, 4)
- (5, 5, 5)
- (4, 6, 5)
- (4, 5, 6)
- (3, 6, 6)

### Step 3: Count the number of distinct arrangements for each combination.
Now we need to account for the arrangements of these combinations:
- (6, 6, 3) can be arranged in \( \frac{3!}{2!} = 3 \) ways.
- (6, 5, 4) can be arranged in \( 3! = 6 \) ways.
- (5, 5, 5) can be arranged in \( \frac{3!}{3!} = 1 \) way.
- (4, 6, 5) can be arranged in \( 3! = 6 \) ways.
- (3, 6, 6) can be arranged in \( \frac{3!}{2!} = 3 \) ways.

Adding these arrangements gives us:
\[ 3 + 6 + 1 + 6 + 3 = 19 \text{ favorable outcomes} \]

### Step 4: Calculate the probability.
The probability \( P \) of getting a sum of 15 when throwing three dice is given by the ratio of favorable outcomes to total outcomes:
\[ P = \frac{\text{Number of Favorable Outcomes}}{\text{Total Outcomes}} = \frac{19}{216} \]

### Final Answer:
The probability that the sum of the three numbers that turn up is 15 is:
\[ \frac{19}{216} \]