There are 6 multiple choice questions in a paper each having 4 options of which only one is correct. In how many ways a person can solve exactly four correct, if he attempted all 6 questions.

Options

A. 134
B. 135
C. 136
D. 137

Answer

To solve the problem of how many ways a person can solve exactly four questions correctly out of six multiple choice questions, we can break it down into a series of steps.

### Step-by-Step Solution:

1. **Identify the total number of questions and options**:
- We have 6 questions, each with 4 options (A, B, C, D), but only one option is correct for each question.

2. **Choose the questions to answer correctly**:
- We need to select 4 questions out of the 6 to answer correctly. The number of ways to choose 4 questions from 6 can be calculated using the combination formula:
\[ \binom{6}{4} \]

3. **Calculate \(\binom{6}{4}\)**:
- Using the combination formula:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
- For our case:
\[ \binom{6}{4} = \frac{6!}{4! \cdot 2!} = \frac{6 \times 5}{2 \times 1} = 15 \]

4. **Choose the questions to answer incorrectly**:
- Since we are answering 4 questions correctly, we will have 2 questions that are answered incorrectly. For each of these incorrectly answered questions, there are 3 wrong options (since there are 4 options total and only 1 is correct).

5. **Calculate the number of ways to answer the 2 questions incorrectly**:
- For each of the 2 questions, there are 3 choices (the wrong answers). Therefore, the total number of ways to answer the 2 questions incorrectly is:
\[ 3 \times 3 = 3^2 = 9 \]

6. **Combine the results**:
- The total number of ways to solve exactly 4 questions correctly and 2 questions incorrectly is the product of the number of ways to choose the correct questions and the number of ways to choose the incorrect answers:
\[ \text{Total Ways} = \binom{6}{4} \times 3^2 = 15 \times 9 = 135 \]

### Final Answer:
Thus, the total number of ways a person can solve exactly four questions correctly is **135**.