There is a question paper consisting of 15 questions. Each question has an internal choice of 3 options. In how many ways can a student attempt one or more questions of the given fifteen questions in the paper?

Options

A. `3^7`
B. `3^8`
C. `3^(15)`
D. `3^(15)-1`

Answer

To solve the problem, we need to determine the number of ways a student can attempt one or more questions from a question paper consisting of 15 questions, each with 3 options (one correct answer and two incorrect answers).

### Step-by-Step Solution:

1. **Understanding the Choices for Each Question:**
Each of the 15 questions has 3 options:
- The student can choose one of the three options (attempt the question).
- The student can choose not to attempt the question (leave it unanswered).

Therefore, for each question, there are a total of 3 choices (attempt option 1, attempt option 2, attempt option 3).

2. **Calculating Total Choices for All Questions:**
Since there are 15 questions and each question has 3 choices, the total number of ways to respond to all questions is given by:
\[ \text{Total choices} = 3^{15} \]

3. **Excluding the Case Where No Questions Are Attempted:**
The total choices calculated above include the scenario where the student does not attempt any questions at all (i.e., chooses to leave all questions unanswered). Since we want to count only the cases where the student attempts at least one question, we need to subtract this one case.

Thus, the number of ways to attempt one or more questions is:
\[ \text{Ways to attempt at least one question} = 3^{15} - 1 \]

4. **Final Answer:**
Therefore, the total number of ways a student can attempt one or more questions is:
\[ 3^{15} - 1 \]