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

Answer: (d) 9

Fix the hundreds digit and count the ways the last two digits can make up the rest: sums of 3, 2 and 1 give 4, 3 and 2 numbers. 1000 does not qualify, so the total is 9.

1. 7ab: a + b = 3 → 703, 712, 721, 730 (4 numbers).
2. 8ab: a + b = 2 → 802, 811, 820 (3 numbers).
3. 9ab: a + b = 1 → 901, 910 (2 numbers).
4. 1000 has digit sum 1, so it does not count. Total = 4 + 3 + 2 = 9.

Remember · Two digits adding to k (k ≤ 9) can be chosen in k + 1 ways.

Question and answer: UPSC's official GS Paper II (2021, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). · Report an error