CSAT Solved Papers/ 2022/Q54

2022 CSAT — Q54

Quant Statement validity 2.5 marks Hard

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:

The $4$ -digit least value of $x$ is $1332$ .
The $3$ -digit greatest value of $x$ is $888$ .

Which of the above statements is/are correct?

A 1 only Answer
B 2 only
C Both 1 and 2
D Neither 1 nor 2

Worked rationale

There are $3! = 6$ numbers; each digit lands in each place exactly $2$ times. So

x = 2(A+B+C)\cdot(100 + 10 + 1) = 222\,(A+B+C).

Let $S = A+B+C$ . With distinct non-zero digits, the smallest sum is $S = 1+2+3 = 6$ , so the smallest possible $x$ is

x_{\min} = 222 \times 6 = 1332.

Statement 1: the least value of $x$ is $1332$ , which is indeed $4$ -digit. Since $x = 222S \ge 1332$ for every admissible triple, $1332$ is both the least value and a $4$ -digit number. Correct.

Statement 2: a $3$ -digit $x$ would need $222S < 1000$ , i.e. $S \le 4$ . But $S \ge 6$ always, so $x$ is never $3$ -digit — no $3$ -digit value of $x$ exists, let alone $888$ ( $= 222\times 4$ , which needs the impossible $S=4$ ). Incorrect.

Answer: (a) 1 only.

Why the other options miss

B

missed a case: computes $888 = 222 \times 4$ and forgets $S = 4$ is impossible for three distinct non-zero digits.
C

missed a case: accepts both numeric coincidences without checking the feasible range of $S$ .
D

wrong formula: derives a wrong place-value coefficient (not $222$ ), rejecting the true Statement 1.

Specialist insight

The engine is the identity $x = 222\,(A+B+C)$ . Everything then turns on the range of $S$ : distinct non-zero digits give $6 \le S \le 24$ , so $x \in \{1332, \dots, 5328\}$ — always $4$ -digit. Statement 2 is a trap built on a value ( $888$ ) that would require $S = 4$ , which the constraints forbid. Always pin the attainable range before judging an extremum claim.

The trap, in one line

$x = 222S$ with $S \ge 6 \Rightarrow x \ge 1332$ always; no $3$ -digit $x$ exists, so only (1) holds $=$ (a).

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