# Q. 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

b. 2 only

c. Both 1 and 2

d. Neither 1 nor 2

Correct Answer: a. 1 only

#### Question from UPSC Prelims 2022 CSAT Paper

**Explanation : **

## Sum of all 3-digit numbers formed by distinct non-zero digits (A, B, C)

Let’s consider all possible 3-digit numbers formed by A, B and C without repetition: ABC, ACB, BAC, BCA, CAB and CBA. The sum of these numbers is x = ABC + ACB + BAC + BCA + CAB + CBA.

We can rewrite x as: x = 100A + 10B + C + 100A + 10C + B + 100B + 10A + C + 100B+10C+A+100C+10A+B+100C+10B+A.

By grouping the terms we get: x = (222)(A+B+C). Since A,B,C are distinct non-zero digits the minimum value for A+B+C is when A=1,B=2,C=3 so the minimum value for x is (222)(1+2+3)=1332. Therefore statement 1 is correct.

Statement 2: The 3-digit greatest value of x is 888: False, as the least value of x is 1332.