UPSC Prelims 2010 Q28: Science & Technology – Quantitative Aptitude
A question paper had ten questions. Each question could only be answered as True (T) or
False (F). Each candidate answered all the questions. Yet, no two candidates wrote the answers in
an identical sequence. How many different sequences of answers are possible?
- (a) 20
- (b) 40
- (c) 512
- (d) 1024
Answer: (d) 1024
Explanation
Counting unique True/False answer sequences
Each of 10 questions can independently be answered True or False.
| Step | Calculation |
|---|---|
| Choices per question | 2 (True or False) |
| Number of questions | 10 |
| Total possible sequences | 2^10 = 1024 |
Why it's a simple power calculation:
| Point | Detail |
|---|---|
| Independent choices | Each question's answer doesn't affect others |
| Formula | (choices per question)^(number of questions) |
| Result | 2^10 = 1024 unique sequences |
Answer: (d) 1024.
Why each option is right or wrong
- (a) Incorrect. 20 is far too small for 10 independent binary choices.
- (b) Incorrect. 40 also massively undercounts the actual number of possible sequences.
- (c) Incorrect. 512 would be 2^9, one question short of the actual 10-question count.
- (d) CORRECT. With 10 independent True/False questions, the total number of sequences is 2^10 = 1024.
Key point
For 10 True/False questions, each independently answerable, the total number of distinct sequences is 2^10 = 1024.
Remember
10 True/False questions → 2^10 = 1024 unique possible answer sequences.
Common mistakes
- Using 2^9 (512) instead of the correct 2^10, miscounting the number of questions.
- Trying to add rather than multiply/exponentiate the choices across questions.
- Confusing this with a smaller, unrelated combinatorial calculation.
Exam tip
n independent binary choices = 2^n total sequences. For 10 True/False questions: 2^10 = 1024.
Sources
- NCERT Class XI Mathematics — Permutations and Combinations
- R. S. Aggarwal — Quantitative Aptitude for Competitive Examinations (Permutations and Combinations)