Integers are listed from 700 to 1000. In how many integers is the sum of the digits 10 ?

(a) 6
(b) 7
(c) 8
(d) 9 ✓ UPSC's answer

Why the answer is (d)

• Integers from 700 to 1000 include 700–999 and 1000; 1000 has digit sum 1, so it is not counted.

• For 700–799, hundreds digit is 7, so tens + units must be 3; the pairs are (0,3), (1,2), (2,1), (3,0), giving 703, 712, 721, 730: 4 numbers.

• For 800–899, hundreds digit is 8, so tens + units must be 2; the pairs are (0,2), (1,1), (2,0), giving 802, 811, 820: 3 numbers.

• For 900–999, hundreds digit is 9, so tens + units must be 1; the pairs are (0,1), (1,0), giving 901, 910: 2 numbers.

• Total = 4 + 3 + 2 = 9, so option (d) is correct.

Why the other options are wrong

(a) 6: Option (a) is 3 short of the total 9 obtained from 4 numbers in the 700s, 3 in the 800s, and 2 in the 900s.
(b) 7: Option (b) is 2 short of the total 9 obtained from 4 numbers in the 700s, 3 in the 800s, and 2 in the 900s.
(c) 8: Option (c) is 1 short of the total 9 obtained from 4 numbers in the 700s, 3 in the 800s, and 2 in the 900s.

Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2021, held on 10 October 2021. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.