The number of times the digit 5 will appear while writing the integers from 1 to 1000 is

(a) 269
(b) 271
(c) 300 ✓ UPSC's answer
(d) 302

Why the answer is (c)

• To count the occurrences of the digit 5 in numbers from 1 to 1000, we analyze each place value (units, tens, hundreds) separately.

• In the units place, the digit 5 appears once in every block of 10 numbers (e.g., 5, 15, 25...), resulting in 100 occurrences in the range 1-999.

• In the tens place, the digit 5 appears 10 times in every block of 100 numbers (e.g., 50-59), resulting in 100 occurrences in the range 1-999.

• In the hundreds place, the digit 5 appears 100 times in the range 500-599.

• The number 1000 does not contain the digit 5, so the total count is 100 + 100 + 100 = 300.

Why the other options are wrong

(a) 269: 269 is incorrect because it undercounts the total occurrences by 31, likely due to miscalculating the frequency in specific place values.
(b) 271: 271 is incorrect because it undercounts the total occurrences by 29, failing to account for the full distribution of the digit 5 across all positions.
(d) 302: 302 is incorrect because it overcounts the total occurrences by 2, possibly by erroneously including the digit 5 in the number 1000 or double-counting specific instances.

Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2019, held on 2 June 2019. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.