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 find the probability that the 6th person asked is the 2nd person to know the answer among 12 people, where exactly 3 out of the 12 know the answer, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
We need to find the probability that the 6th person asked is the 2nd person who knows the answer. This means that among the first 5 people asked, exactly 1 person must know the answer, and the 6th person must be the 2nd person who knows the answer.

2. **Choosing the People**:
- We have 3 people who know the answer (let's call them A, B, and C).
- We have 9 people who do not know the answer (let's call them D1, D2, ..., D9).
- We need to choose 1 person from the 3 who knows the answer to be among the first 5 asked (this can be done in \( \binom{3}{1} = 3 \) ways).
- We also need to choose 4 people from the 9 who do not know the answer (this can be done in \( \binom{9}{4} \) ways).

3. **Calculating the Total Ways**:
The total number of ways to choose 5 people from 12 is \( \binom{12}{5} \).

4. **Arranging the Chosen People**:
The arrangement of the chosen 5 people (1 who knows the answer and 4 who do not) can occur in any order. The 6th person (who knows the answer) must be placed after these 5.

5. **Calculating the Probability**:
The probability that the 6th person asked is the 2nd person to know the answer can be calculated as follows:
\[ P = \frac{\text{Ways to choose 1 from 3 who know the answer} \times \text{Ways to choose 4 from 9 who do not know the answer}}{\text{Total ways to choose 5 from 12}} \times \text{Probability of the 6th person knowing the answer} \]

This can be expressed mathematically as:
\[ P = \frac{\binom{3}{1} \cdot \binom{9}{4}}{\binom{12}{5}} \cdot \frac{2}{7} \]

Here, \( \frac{2}{7} \) is the probability that the 6th person is one of the 2 remaining who know the answer after 1 has been chosen among the first 5.

6. **Calculating the Combinations**:
- \( \binom{3}{1} = 3 \)
- \( \binom{9}{4} = \frac{9!}{4!(9-4)!} = \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2 \times 1} = 126 \)
- \( \binom{12}{5} = \frac{12!}{5!(12-5)!} = \frac{12 \times 11 \times 10 \times 9 \times 8}{5 \times 4 \times 3 \times 2 \times 1} = 792 \)

7. **Final Calculation**:
Substituting the values into the probability formula:
\[ P = \frac{3 \cdot 126}{792} \cdot \frac{2}{7} \]
\[ = \frac{378}{792} \cdot \frac{2}{7} \]
\[ = \frac{378 \cdot 2}{792 \cdot 7} = \frac{756}{5544} = \frac{3}{22} \]

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