Three dice are thrown together. The probability that the sum of the numbers appearing on them is 9, is

Options

A. `1//216`
B. `1//72`
C. `5//108`
D. `1//18`

Answer

To find the probability that the sum of the numbers appearing on three thrown dice is 9, we can follow these steps:

### Step 1: Determine the Total Outcomes
When three dice are thrown, each die has 6 faces. Therefore, the total number of outcomes when throwing three dice is:
\[ 6^3 = 216 \]

### Step 2: Find the Favorable Outcomes
Next, we need to find the combinations of the numbers on the three dice that add up to 9. We can list the combinations systematically:

1. **(1, 2, 6)**
2. **(1, 3, 5)**
3. **(1, 4, 4)**
4. **(2, 2, 5)**
5. **(2, 3, 4)**
6. **(3, 3, 3)**

Now, we need to determine how many distinct permutations each of these combinations can produce:

- **(1, 2, 6)**: All numbers are different. The number of permutations = \(3! = 6\).
- **(1, 3, 5)**: All numbers are different. The number of permutations = \(3! = 6\).
- **(1, 4, 4)**: Two numbers are the same. The number of permutations = \(\frac{3!}{2!} = 3\).
- **(2, 2, 5)**: Two numbers are the same. The number of permutations = \(\frac{3!}{2!} = 3\).
- **(2, 3, 4)**: All numbers are different. The number of permutations = \(3! = 6\).
- **(3, 3, 3)**: All numbers are the same. The number of permutations = 1.

### Step 3: Calculate the Total Favorable Outcomes
Now, we add the number of permutations for each combination:
\[ 6 + 6 + 3 + 3 + 6 + 1 = 25 \]

### Step 4: Calculate the Probability
The probability \(P\) that the sum of the numbers appearing on the dice is 9 is given by the ratio of the number of favorable outcomes to the total outcomes:
\[ P = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{25}{216} \]

### Final Answer
Thus, the probability that the sum of the numbers appearing on the three dice is 9 is:
\[ \frac{25}{216} \]

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