UPSC Prelims 2006 Q92: Science & Technology – Quantitative Aptitude
In a question paper, there are four multiple - choice questions. Each question has five choices
with only one choice for its correct answer. What is the total number of ways in which a
candidate will not get all the four answers correct?
- (a) 19
- (b) 120
- (c) 624
- (d) 1024
Answer: (c) 624
Explanation
Ways of not getting all four answers correct
Each question has 5 choices and only one is correct, so every question can be answered in 5 ways.
| Step | Calculation |
|---|---|
| Total ways to answer four questions | 5 x 5 x 5 x 5 = 625 |
| Ways of getting all four correct | 1 |
| Ways of not getting all four correct | 625 - 1 = 624 |
So the candidate can fail to get all four answers right in 624 ways.
Answer: (c) 624.
Why each option is right or wrong
- (a) Incorrect. 19 is far below the number of wrong combinations.
- (b) Incorrect. 120 is 5 factorial and does not apply here.
- (c) CORRECT. 5^4 - 1 = 625 - 1 = 624.
- (d) Incorrect. 1024 is 4^5, which swaps the base and exponent.
Key point
Total answer patterns are 5^4 = 625, and only one is fully correct, so 624 ways are not all correct.
Remember
5^4 - 1 = 624 ways of not getting all four correct.
Common mistakes
- Using 4^5 instead of the correct 5^4 count.
- Forgetting to subtract the one fully correct pattern.
- Counting only arrangements rather than choices per question.
Exam tip
n choices for each of r questions: n^r patterns; subtract 1 for the all-correct one.
Sources
- NCERT Class XI Mathematics — Permutations and Combinations
- R. S. Aggarwal — Quantitative Aptitude for Competitive Examinations (Permutations and Combinations)