In an examination, there are 3 multi-choice questions and each question has 4 alternatives. If a student is declared pass only when he attempts all question correctly, then number of ways in which he can fail is

Options

A. 1
B. 12
C. 27
D. 63

Answer

To solve the problem, we need to determine the number of ways a student can fail the examination given the conditions stated. Here’s a step-by-step breakdown of the solution:

### Step 1: Determine the total number of ways to answer the questions.
Each of the 3 multiple-choice questions has 4 alternatives. Therefore, the total number of ways to answer all questions can be calculated using the formula for combinations:

\[ \text{Total ways} = 4 \times 4 \times 4 = 4^3 = 64 \]

### Step 2: Determine the number of ways to pass the examination.
A student passes the examination only if they answer all questions correctly. Since each question has only one correct answer, the number of ways to pass is:

\[ \text{Ways to pass} = 1 \times 1 \times 1 = 1 \]

### Step 3: Calculate the number of ways to fail the examination.
To find the number of ways a student can fail, we subtract the number of ways to pass from the total number of ways to answer the questions:

\[ \text{Ways to fail} = \text{Total ways} - \text{Ways to pass} = 64 - 1 = 63 \]

### Conclusion
Thus, the number of ways in which the student can fail the examination is **63**.

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