In a 'multiple choice question' test there are eight questions. Each question has four alternative of which only one is correct. IF a candidate answers all the questions by choosing one answer for each question, then the number of ways to get exactly 4 correct answer is

Options

A. 70
B. 2835
C. 5670
D. none of these

Answer

To solve the problem of finding the number of ways to get exactly 4 correct answers out of 8 multiple choice questions, we can break down the solution into several steps.

### Step 1: Choose 4 questions to answer correctly
We need to select 4 questions out of the 8 to answer correctly. This can be calculated using the combination formula \( \binom{n}{r} \), where \( n \) is the total number of questions and \( r \) is the number of questions we want to choose.

\[ \text{Number of ways to choose 4 questions from 8} = \binom{8}{4} \]

### Step 2: Calculate \( \binom{8}{4} \)
Using the formula for combinations:

\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]

We can substitute \( n = 8 \) and \( r = 4 \):

\[ \binom{8}{4} = \frac{8!}{4!(8-4)!} = \frac{8!}{4!4!} \]

Calculating the factorials:

\[ = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = \frac{1680}{24} = 70 \]

### Step 3: Choose answers for the remaining 4 questions
For the remaining 4 questions, since we want them to be incorrect, each of these questions has 3 incorrect options (since there are 4 options total and only one is correct). Therefore, for each of the 4 questions, there are 3 ways to choose an incorrect answer.

\[ \text{Number of ways to answer incorrectly for 4 questions} = 3^4 \]

### Step 4: Calculate \( 3^4 \)
Calculating \( 3^4 \):

\[ 3^4 = 81 \]

### Step 5: Combine the results
Now, we multiply the number of ways to choose the correct answers by the number of ways to choose the incorrect answers:

\[ \text{Total ways} = \binom{8}{4} \times 3^4 = 70 \times 81 \]

### Step 6: Calculate the final result
Calculating \( 70 \times 81 \):

\[ 70 \times 81 = 5670 \]

### Final Answer
Thus, the total number of ways to get exactly 4 correct answers is:

\[ \boxed{5670} \]

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