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?

(a) 1 only ✓ UPSC's answer
(b) 2 only
(c) Both 1 and 2
(d) Neither 1 nor 2

Why the answer is (a)

• For any three distinct non-zero digits A, B, and C, there are 6 possible 3-digit numbers formed without repetition: ABC, ACB, BAC, BCA, CAB, and CBA.

• The sum of these numbers is calculated as 222(A + B + C), because each digit appears in the hundreds, tens, and units place exactly twice.

• To find the least value of x, we must minimize the sum (A + B + C). The smallest distinct non-zero digits are 1, 2, and 3, giving a sum of 6.

• The minimum value of x is 222 * 6 = 1332, which is a 4-digit number, making Statement 1 correct.

• To find the greatest value of x, we maximize (A + B + C) using the largest distinct non-zero digits 9, 8, and 7, giving a sum of 24.

• The maximum value of x is 222 * 24 = 5328, which is a 4-digit number, not a 3-digit number, making Statement 2 incorrect.

Why the other options are wrong

(b) 2 only: Statement 2 is incorrect because the greatest value of x is 5328, which is a 4-digit number, not a 3-digit number.
(c) Both 1 and 2: Statement 2 is incorrect because the maximum sum of the digits (9+8+7=24) results in x=5328, which exceeds the 3-digit limit.
(d) Neither 1 nor 2: Statement 1 is correct because the minimum sum of distinct non-zero digits (1+2+3=6) results in x=1332, which is indeed the least 4-digit value.

Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2022, held on 5 June 2022. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.