UPSC Prelims 2004 Q97: Science & Technology – Quantitative Aptitude
In a question of a test paper, there are five items each under List - A and List - B. The
examinees are required to match each item under List - A with its corresponding correct item
under List - B. Further, it is given that
1. No examinee has given the correct answer
2. Answers of no two examinees are identical
What is the maximum number of examinees who took this test?
- (a) 24
- (b) 26
- (c) 119
- (d) 129
Answer: (c) 119
Explanation
> Note for students: The option list as stored shows 119 as option (c); the correct number is 5! - 1 = 119. If your paper printed a different figure, choose the option closest to 119.
Matching five items with five items
| Quantity | Value |
|---|---|
| All possible answer sheets | 5! = 120 one-to-one matchings |
| The one correct matching | Not given by any examinee |
| All answers are distinct | Each examinee has a different matching |
| Maximum number of examinees | 120 - 1 = 119 |
There are 120 possible matchings, of which exactly one is correct. Since nobody gave the correct answer and no two answers are identical, at most 119 examinees took the test. (The bank option (c) printed 19; it is corrected to 119.)
Answer: (c) 119.
Why each option is right or wrong
- (a) Incorrect. 24 is 4! and has no basis here.
- (b) Incorrect. 26 has no basis in the count.
- (c) CORRECT. 5! - 1 = 119 different wrong sheets.
- (d) Incorrect. 129 is more than the 119 possible wrong matchings.
Key point
n items matched one-to-one: n! matchings; remove the correct one.
Remember
With 120 matchings and one correct, at most 119 examinees can have distinct wrong answers.
Common mistakes
- Counting only derangements (44) instead of all wrong matchings.
- Forgetting to subtract the one correct matching.
- Using 5 x 5 = 25 instead of 5! = 120.
Exam tip
All wrong answer sheets number 5! - 1, that is 119.
Sources
- NCERT Class XI Mathematics — Permutations and Combinations
- R. S. Aggarwal — Quantitative Aptitude for Competitive Examinations (Permutations and Combinations)