12 people are asked questions in succession in a random order and exactly 3 out of 12 people know the answer. The probability that the `6^("th")` person asked is the `2^("nd")` person to know the answer, is

Options

A. `(10)/(21)`
B. `(3)/(22)`
C. `(7)/(11)`
D. `(5)/(12)`

Answer

To solve the problem, we need to find the probability that the 6th person asked is the 2nd person to know the answer, given that there are 12 people and exactly 3 of them know the answer.

### Step-by-Step Solution:

1. **Identify the Total Number of People and Knowledgeable Individuals**:
- Total people = 12
- People who know the answer = 3
- People who do not know the answer = 12 - 3 = 9

2. **Define the Event**:
- We want the 6th person asked to be the 2nd person who knows the answer.

3. **Conditions for the 6th Person**:
- For the 6th person to be the 2nd person who knows the answer, exactly one of the first five people asked must know the answer.

4. **Calculate the Ways to Choose the First Five People**:
- We need to choose 1 person who knows the answer from the 3 knowledgeable people. This can be done in \( \binom{3}{1} \) ways.
- We also need to choose 4 people who do not know the answer from the 9 who do not know. This can be done in \( \binom{9}{4} \) ways.

5. **Calculate the Total Arrangements of the First Five People**:
- The total arrangements of these 5 people (1 who knows and 4 who do not) can be calculated as \( 5! \).

6. **Calculate the Total Arrangements Including the 6th Person**:
- The 6th person (who knows the answer) can be added to the arrangement. The remaining 6 people (2 who know and 5 who do not) can be arranged in \( 7! \) ways.

7. **Calculate the Total Outcomes**:
- The total number of ways to arrange all 12 people is \( 12! \).

8. **Combine Everything to Find the Probability**:
- The probability \( P \) that the 6th person is the 2nd person to know the answer is given by:
\[ P = \frac{\text{Ways to arrange the first 5 with 1 knowledgeable}}{\text{Total arrangements of all 12}} \]
\[ P = \frac{\binom{3}{1} \cdot \binom{9}{4} \cdot 5! \cdot 7!}{12!} \]

9. **Simplify the Expression**:
- Substitute the values:
\[ \binom{3}{1} = 3, \quad \binom{9}{4} = \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2 \times 1} = 126 \]
- Thus, the probability becomes:
\[ P = \frac{3 \cdot 126 \cdot 120 \cdot 5040}{479001600} \]

10. **Final Calculation**:
- After simplification, we find that:
\[ P = \frac{3 \cdot 126 \cdot 120 \cdot 5040}{12!} = \frac{3 \cdot 126 \cdot 120}{11 \cdot 10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1} \]
- This simplifies to \( \frac{3}{22} \).

### Final Answer:
The probability that the 6th person asked is the 2nd person to know the answer is \( \frac{3}{22} \).