The consists of 6 multiple choice questions, each having 4 alternative answers of wihc only one is correct. The number of ways, in which a canditate answers all six questions such that exactly four of the answers are correct, is _______.

Answer

To solve the problem of determining the number of ways a candidate can answer 6 multiple choice questions such that exactly 4 answers are correct, we can break the solution down into several steps.

### Step-by-Step Solution:

1. **Identify the Total Questions and Correct Answers**:
We have a total of 6 questions, and we want exactly 4 of them to be answered correctly.

2. **Choose Which Questions are Correct**:
We need to select 4 questions out of the 6 to be the correct ones. This can be done using the combination formula:
\[ \binom{6}{4} \]
The value of \(\binom{6}{4}\) can also be calculated as \(\binom{6}{2}\) since \(\binom{n}{k} = \binom{n}{n-k}\).

3. **Calculate the Combinations**:
\[ \binom{6}{4} = \frac{6!}{4!(6-4)!} = \frac{6!}{4! \cdot 2!} = \frac{6 \times 5}{2 \times 1} = 15 \]

4. **Determine the Ways to Answer Correctly**:
For each of the 4 questions that are correct, there is only 1 way to answer correctly (since there is only one correct option out of 4). Therefore, for 4 correct answers, the number of ways is:
\[ 1^4 = 1 \]

5. **Determine the Ways to Answer Incorrectly**:
For the 2 questions that are incorrect, there are 3 wrong options for each question (since there are 4 options in total and only 1 is correct). Thus, the number of ways to answer incorrectly for 2 questions is:
\[ 3^2 = 9 \]

6. **Combine the Results**:
Now, we combine the results from the previous steps to find the total number of ways to answer the questions:
\[ \text{Total Ways} = \binom{6}{4} \times 1^4 \times 3^2 = 15 \times 1 \times 9 = 135 \]

### Final Answer:
The number of ways in which a candidate answers all six questions such that exactly four of the answers are correct is **135**.

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