UPSC Prelims 2004 Q136: Science & Technology – Quantitative Aptitude
- (a) 120
- (b) 210
- (c) 365
- (d) 405
Answer: (d) 405
Explanation
> Note for students: The factual count is 325, which is not among the printed options (120, 210, 365, 405). The published key is (d) 405. Choose (d) or skip it to avoid a negative mark.
Counting three-digit even numbers with a 7-9 condition
| Case | Count |
|---|---|
| Middle digit 9, preceded by 7 | Numbers 79x with x even: 5 |
| Middle digit 7 | Impossible: 7 would have to be followed by 9 in the last place, which is odd |
| Middle digit neither 7 nor 9 | First digit 1-9 except 7 (8 ways), middle digit from 0-9 except 7 and 9 (8 ways), last digit even (5 ways): 8 x 8 x 5 = 320 |
| Total | 5 + 320 = 325 |
Every 9 that follows another digit must follow a 7, and every 7 that is followed by a digit must be followed by a 9. Counting the allowed cases gives 325, which is not one of the printed options (120, 210, 365, 405). The published key is (d) 405; it is kept as the key, but the factual count is 325 and the question has no exactly correct option.
Answer: (d) 405 (published key; the factual count is 325, not among the options).
Why each option is right or wrong
- (a) Incorrect. 120 is far below the number of valid numbers.
- (b) Incorrect. 210 is far below the number of valid numbers.
- (c) Incorrect. 365 counts only the clause on 7 and omits the clause on 9.
- (d) CORRECT. This is the published key, although the factual count is 325.
Key point
Split by the middle digit, and apply the condition to each adjacent pair.
Remember
The factual count of such three-digit even numbers is 325.
Common mistakes
- Counting only the clause on 9 or only the clause on 7.
- Forgetting that 9 may stand in the first place.
- Ignoring that the number must be even.
Exam tip
Test the conditions by brute force on adjacent pairs: it gives 325.
Sources
- NCERT Class XI Mathematics — Permutations and Combinations
- R. S. Aggarwal — Quantitative Aptitude for Competitive Examinations (Permutations and Combinations)