Let A, B and C represent distinct non-zero digits. Suppose x is the sum of all possible 3-digit numbers formed by A, B and C without repetition.
Consider the following statements:
- 1.The 4-digit least value of x is 1332.
- 2.The 3-digit greatest value of x is 888.
Which of the above statements is/are correct?
Answer & explanation
Answer: (a) 1 only
The six numbers use each digit twice in every place, so x = 222 × (A + B + C). The least digit sum is 1 + 2 + 3 = 6, giving x = 1332, so x is always at least 1332 and can never be a 3-digit number.
- The six numbers formed by A, B, C place each digit twice in the hundreds, tens and units places, so x = 2 × (A + B + C) × 111 = 222 × (A + B + C).
- The smallest possible sum of three distinct non-zero digits is 1 + 2 + 3 = 6, so the least x is 222 × 6 = 1332, a 4-digit number. Statement 1 is correct.
- A 3-digit x would need 222 × (A + B + C) < 1000, i.e. a digit sum of 4 or less — impossible. So x is never 3-digit, and 888 (digit sum 4) cannot occur. Statement 2 is incorrect.
- Check: 123 + 132 + 213 + 231 + 312 + 321 = 1332.
- ✓ 1. Digit sum 6 is the minimum, giving x = 222 × 6 = 1332.
- ✗ 2. x is always a multiple of 222 with digit sum at least 6, so it is never below 1332; 888 would need digit sum 4.
Remember · Sum of all permutations of three distinct digits = 222 × (sum of the digits).
Question and answer: UPSC's official GS Paper II (2022, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). ·