What is the sum of all digits which appear in all the integers from 10 to 100 ?

(a) 855
(b) 856 ✓ UPSC's answer
(c) 910
(d) 911

Why the answer is (b)

• The integers from 10 to 100 consist of 91 numbers: 90 two-digit numbers (10-99) and one three-digit number (100).

• For the number 100, the sum of digits is 1 + 0 + 0 = 1.

• For the 90 two-digit numbers (10-99), the sum of the units digits is calculated by noting that digits 0-9 each appear 9 times in the units place (e.g., 0 in 10, 20...90), so the sum is 9 * (0+1+...+9) = 9 * 45 = 405.

• The sum of the tens digits is calculated by noting that digits 1-9 each appear 10 times in the tens place (e.g., 1 in 10-19), so the sum is 10 * (1+2+...+9) = 10 * 45 = 450.

• The total sum for the two-digit numbers is 405 + 450 = 855.

• Adding the sum from the number 100 gives a grand total of 855 + 1 = 856, which corresponds to option (b).

Why the other options are wrong

(a) 855: Option (a) is 855, which is the sum of digits for only the two-digit numbers (10-99), excluding the digit sum of 100.
(c) 910: Option (c) is 910, which is an incorrect calculation likely resulting from miscounting the frequency of digits or including an extra 55.
(d) 911: Option (d) is 911, which is an incorrect calculation likely resulting from miscounting the frequency of digits or including an extra 56.

Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2023, held on 28 May 2023. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.