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

Correct Answer: Option B

Why Option B is correct: In the study of numerical combinations and basic arithmetic—a core component of the UPSC CSAT (Paper II) Basic Numeracy syllabus—counting digit occurrences requires systematic enumeration. Two-digit positive integers strictly range from 10 to 99. To determine how many times the digit '5' appears, we analyze its occurrences in the tens place and the units place separately.

1. Tens Place: The digit '5' appears in the tens place for the integers 50, 51, 52, 53, 54, 55, 56, 57, 58, and 59. This accounts for exactly 10 occurrences.
2. Units Place: The digit '5' appears in the units place for the integers 15, 25, 35, 45, 55, 65, 75, 85, and 95. Note that '05' is a single-digit integer and is thus excluded. This accounts for exactly 9 occurrences. Summing both positional counts (10 + 9) yields a total of 19 occurrences.

Why other options are incorrect:

• Option A (18) is incorrect: This is a common pitfall where an aspirant counts the integers containing the digit '5' rather than the occurrences of the digit itself, effectively missing the second '5' in '55'.
• Option C (20) is incorrect: This error typically occurs if one incorrectly includes the single-digit number '5' (mentally calculating the range 1 to 99 instead of 10 to 99), which falsely adds a 10th unit-place occurrence.
• Option D (21) is incorrect: This is the result of an arbitrary over-calculation or double-counting boundaries in mental math.

Concluding Takeaway: When calculating digit frequencies for CSAT problems, evaluate each place value (units, tens) independently and sum the results. Always distinguish carefully between 'how many integers contain a digit' and 'how many times the digit appears.'