UPSC Prelims 2009 Q96: Science & Technology – Quantitative Aptitude
How many numbers lie between 300 and 500 in which 4 comes only one time?
- (a) 99
- (b) 100
- (c) 110
- (d) 120
Answer: (a) 99
Explanation
Counting numbers between 300 and 500 where digit "4" appears exactly once
We need three-digit numbers in the 300-499 range where the digit 4 appears exactly one time.
| Case | Calculation |
|---|---|
| Hundreds digit = 3 (300-399) | The single '4' must appear in the tens or units place (not both, and hundreds digit is fixed at 3, not 4): Tens=4,Units≠4 gives 9 options; Units=4,Tens≠4 gives 9 options → 18 total |
| Hundreds digit = 4 (400-499) | The hundreds digit itself is the one '4'; tens and units must both be non-4 (9 options each): 9 × 9 = 81 |
| Total | 18 + 81 = 99 |
Why we split into two cases:
| Point | Detail |
|---|---|
| 300s range | '4' can only appear in tens or units |
| 400s range | Hundreds digit is already the one '4'; tens/units must avoid it |
Answer: (a) 99.
Why each option is right or wrong
- (a) CORRECT. Adding the 300s case (18) and the 400s case (81) gives 99 numbers with exactly one '4'.
- (b) Incorrect. 100 slightly overcounts the correct total.
- (c) Incorrect. 110 overstates the actual count.
- (d) Incorrect. 120 significantly overstates the actual count.
Key point
Between 300 and 500, numbers with exactly one '4': 18 (from the 300s, where '4' appears once in tens/units) + 81 (from the 400s, where the hundreds digit IS the one '4' and tens/units avoid it) = 99.
Remember
Numbers between 300-500 with digit '4' appearing exactly once: 99 (18 from the 300s + 81 from the 400s).
Common mistakes
- Forgetting to split into two separate cases (300s and 400s) based on where the hundreds digit sits.
- Double-counting or missing the constraint that tens AND units must both avoid '4' in the 400s case.
- Making an arithmetic error while combining the two case totals.
Exam tip
For 'digit appears exactly once' counting problems, split by which position could hold that digit, and carefully apply the 'exactly once' constraint to each case.
Sources
- NCERT Class XI Mathematics — Permutations and Combinations
- R. S. Aggarwal — Quantitative Aptitude for Competitive Examinations (Permutations and Combinations)