Question Details

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?

Options

1 only

2 only

Both 1 and 2

Neither 1 nor 2

Show Answer

Correct Answer :

Option A

1 only

Solution :

The correct option is 1 only.

Let us break down the mathematical derivation step-by-step to understand why this statement is correct and statement 2 is incorrect.

Let $A$ , $B$ , and $C$ be distinct non-zero digits. Thus, $A, B, C \in {1, 2, 3, 4, 5, 6, 7, 8, 9}$ with $A \neq B$ , $B \neq C$ , and $A \neq C$ .

The 3-digit numbers that can be formed using the digits $A$ , $B$ , and $C$ without repetition are:
1. $A B C = 100 A + 10 B + C$
2. $A C B = 100 A + 10 C + B$
3. $B A C = 100 B + 10 A + C$
4. $B C A = 100 B + 10 C + A$
5. $C A B = 100 C + 10 A + B$
6. $C B A = 100 C + 10 B + A$

The sum of all these 3-digit numbers is $x$ . Adding them up, we get:

$x = (100 A + 100 A + 100 B + 100 B + 100 C + 100 C) + (10 B + 10 C + 10 A + 10 C + 10 A + 10 B) + (C + B + C + A + B + A)$

$x = 200 (A + B + C) + 20 (A + B + C) + 2 (A + B + C)$

$x = 222 (A + B + C)$

Now, let us analyze the statements based on the relation $x = 222 (A + B + C)$ .

Statement 1: The 4-digit least value of x is 1332.
To find the least value of $x$ that is a 4-digit number, we need the sum of the digits $A + B + C$ to be as small as possible such that $x \geq 1000$ .
Since $A$ , $B$ , and $C$ are distinct non-zero digits, the smallest possible sum of $A + B + C$ is achieved when we select the three smallest non-zero digits, which are 1, 2, and 3:
$A + B + C = 1 + 2 + 3 = 6$
Substituting this value into the expression for $x$ :
$x = 222 \times 6 = 1332$
This is indeed a 4-digit number, and since $1 + 2 + 3 = 6$ is the absolute minimum possible sum of three distinct non-zero digits, 1332 is the smallest possible value of $x$ (which happens to be a 4-digit number). Therefore, Statement 1 is correct.

Statement 2: The 3-digit greatest value of x is 888.
For $x$ to be a 3-digit number, it must be less than 1000.
We set up the inequality:
$222 (A + B + C) < 1000$
Dividing both sides by 222:
$A + B + C < \frac{1000}{222} \approx 4.5$
Since $A$ , $B$ , and $C$ are distinct non-zero digits, the minimum possible sum they can have is:
$A + B + C = 1 + 2 + 3 = 6$
Since $6$ is not less than $4.5$ , there is no combination of distinct non-zero digits $A, B, C$ that yields a sum less than 6. Consequently, it is impossible for $x$ to be a 3-digit number. The minimum value of $x$ is 1332, which is already a 4-digit number. Therefore, Statement 2 is incorrect.