Arranging digits in rows based on multiplication conditions

Let the 3-digit number in the first row be x. The second row is 2x, and the third row is 3x. Because the third row must be a 3-digit number, 3x < 1000 \implies x < 333.

The digits for x must be chosen from the set \{2, 3, 7, 9\}. Since x < 333, its hundreds digit must be 2 or 3. If the hundreds digit is 3, the tens digit cannot be 7 or 9 because 37x × 3 > 1000.

Test the valid permutations sequentially:

Only exactly 2 combinations survive the strict 9-digit non-repeating constraint.

Answer: (c).