How many times does 5 appear in all two-digit positive integers?

(a) 18
(b) 19 ✓ UPSC's answer
(c) 20
(d) 21

Why the answer is (b)

• Two-digit positive integers range from 10 to 99.

• Count occurrences of 5 in the units place: 15, 25, 35, 45, 55, 65, 75, 85, 95 (9 times).

• Count occurrences of 5 in the tens place: 50, 51, 52, 53, 54, 55, 56, 57, 58, 59 (10 times).

• The number 55 contains the digit 5 in both the tens and units places, so it is counted in both sets above.

• Total count is 9 (units) + 10 (tens) = 19.

• Therefore, the digit 5 appears 19 times in all two-digit positive integers.

Why the other options are wrong

(a) 18: Option (a) is incorrect because it undercounts the total occurrences by 1, likely missing one instance in either the tens or units position.
(c) 20: Option (c) is incorrect because it overcounts the total occurrences by 1, possibly double-counting 55 or including a non-existent case.
(d) 21: Option (d) is incorrect because it overcounts the total occurrences by 2, which does not match the systematic count of 19.

Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2026, held on 24 May 2026. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.