UPSC Prelims 2006 Q136: Science & Technology – Quantitative Aptitude
How many numbers are there in all 6000 to 6999 (both 6000 and 6999 included) having at
least one of their digits repeated?
- (a) 216
- (b) 356
- (c) 496
- (d) 504
Answer: (c) 496
Explanation
Numbers from 6000 to 6999 with at least one repeated digit
Count the complement: numbers with all digits different.
| Step | Calculation |
|---|---|
| Total numbers from 6000 to 6999 | 1000 |
| Numbers with all digits distinct | First digit fixed as 6; the other three digits come from the remaining 9 digits (0 to 9 excluding 6): 9 x 8 x 7 = 504 |
| Numbers with at least one repeated digit | 1000 - 504 = 496 |
Answer: (c) 496.
Why each option is right or wrong
- (a) Incorrect. 216 is not the count of numbers with a repeated digit.
- (b) Incorrect. 356 does not match 1000 - 504.
- (c) CORRECT. 1000 - 504 = 496 numbers have a repeated digit.
- (d) Incorrect. 504 is the count of numbers with all digits distinct, not the repeated ones.
Key point
There are 1000 numbers, of which 9 x 8 x 7 = 504 have all distinct digits, leaving 496 with a repeated digit.
Remember
6000-6999: 1000 total, 504 distinct, so 496 have a repeated digit.
Common mistakes
- Choosing 504, the count of numbers with distinct digits.
- Forgetting that zero is allowed in the last three places.
- Counting the first digit 6 as free rather than fixed.
Exam tip
'At least one repeated' = total minus 'all different'.
Sources
- Arun Sharma — How to Prepare for Quantitative Aptitude for the CAT (Permutations and Combinations)
- R. S. Aggarwal — Quantitative Aptitude for Competitive Examinations (Permutations and Combinations)