An examinee is required to answer six questions out of twelve questions which are divided into two groups each containing six questions and he is not permitted to answer more than four questions from any group. In how many ways can he answer six questions?

Options

A. 750
B. 850
C. 580
D. 570

Answer

To solve the problem of how many ways an examinee can answer six questions out of twelve, divided into two groups (A and B) of six questions each, with the restriction that no more than four questions can be answered from any group, we can break down the solution into several cases.

### Step-by-step Solution:

1. **Identify the Groups**:
- Group A: Questions 1, 2, 3, 4, 5, 6
- Group B: Questions 7, 8, 9, 10, 11, 12

2. **Define the Cases**:
The examinee can answer questions in the following combinations:
- Case 1: 2 questions from Group A and 4 questions from Group B
- Case 2: 3 questions from Group A and 3 questions from Group B
- Case 3: 4 questions from Group A and 2 questions from Group B

3. **Calculate Each Case**:
- **Case 1**:
- Choose 2 questions from Group A:
\[ \text{Number of ways} = \binom{6}{2} = \frac{6!}{2!(6-2)!} = \frac{6 \times 5}{2 \times 1} = 15 \]
- Choose 4 questions from Group B:
\[ \text{Number of ways} = \binom{6}{4} = \binom{6}{2} = 15 \]
- Total for Case 1:
\[ 15 \times 15 = 225 \]

- **Case 2**:
- Choose 3 questions from Group A:
\[ \text{Number of ways} = \binom{6}{3} = \frac{6!}{3!(6-3)!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20 \]
- Choose 3 questions from Group B:
\[ \text{Number of ways} = \binom{6}{3} = 20 \]
- Total for Case 2:
\[ 20 \times 20 = 400 \]

- **Case 3**:
- Choose 4 questions from Group A:
\[ \text{Number of ways} = \binom{6}{4} = 15 \]
- Choose 2 questions from Group B:
\[ \text{Number of ways} = \binom{6}{2} = 15 \]
- Total for Case 3:
\[ 15 \times 15 = 225 \]

4. **Add All Cases Together**:
- Total ways = Case 1 + Case 2 + Case 3
\[ \text{Total} = 225 + 400 + 225 = 850 \]

### Final Answer:
The examinee can answer the questions in **850 different ways**.

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