The digits 1 to 9 are arranged in three rows in such a way that each row contains three digits, and the number formed in the second row is twice the number formed in the first row; and the number formed in the third row is thrice the number formed in the first row. Repetition of digits is not allowed. If only three of the four digits 2, 3, 7 and 9 are allowed to use in the first row, how many such combinations are possible to be arranged in the three rows?

Correct Option (3)

The problem requires arranging the digits 1 to 9 into three distinct three-digit numbers, denoted as Row 1, Row 2, and Row 3, subject to the following conditions:

• Each row must contain three digits.
• The number formed in Row 2 must be twice the number formed in Row 1 (Row 2 = 2 × Row 1).
• The number formed in Row 3 must be thrice the number formed in Row 1 (Row 3 = 3 × Row 1).
• Repetition of digits is not allowed across all three numbers combined; all nine digits (1-9) must be used exactly once.
• Only three of the four digits {2, 3, 7, 9} are allowed to be used in the first row.

To determine the possible values for Row 1, we first establish its upper bound:

• Since Row 3 is a three-digit number, 3 × Row 1 must be less than 1000.
• This implies Row 1 < 1000 / 3, so Row 1 < 333.33.
• Therefore, Row 1 must be a three-digit number less than or equal to 333.

Next, we identify all possible three-digit numbers for Row 1 that can be formed using three distinct digits from the set {2, 3, 7, 9} and are less than or equal to 333:

• Numbers starting with 2: 237, 239, 273, 279, 293, 297.
• Numbers starting with 3: 327, 329. (Numbers like 37x or 39x would exceed 333).

Now, we evaluate each potential Row 1 number by calculating Row 2 and Row 3, and then checking if all nine digits used across the three numbers are unique (i.e., no repetition):

• If Row 1 = 237: Row 2 = 474, Row 3 = 711. Digits used: {2,3,7,4,7,4,7,1,1}. Digits 1, 4, and 7 are repeated. This combination is invalid.
• If Row 1 = 239: Row 2 = 478, Row 3 = 717. Digits used: {2,3,9,4,7,8,7,1,7}. Digit 7 is repeated. This combination is invalid.
• If Row 1 = 273: Row 2 = 546, Row 3 = 819. Digits used: {2,7,3,5,4,6,8,1,9}. All nine digits (1, 2, 3, 4, 5, 6, 7, 8, 9) are unique. This combination is valid.
• If Row 1 = 279: Row 2 = 558, Row 3 = 837. Digits used: {2,7,9,5,5,8,8,3,7}. Digits 5, 7, and 8 are repeated. This combination is invalid.
• If Row 1 = 293: Row 2 = 586, Row 3 = 879. Digits used: {2,9,3,5,8,6,8,7,9}. Digits 8 and 9 are repeated. This combination is invalid.
• If Row 1 = 297: Row 2 = 594, Row 3 = 891. Digits used: {2,9,7,5,9,4,8,9,1}. Digit 9 is repeated. This combination is invalid.
• If Row 1 = 327: Row 2 = 654, Row 3 = 981. Digits used: {3,2,7,6,5,4,9,8,1}. All nine digits (1, 2, 3, 4, 5, 6, 7, 8, 9) are unique. This combination is valid.
• If Row 1 = 329: Row 2 = 658, Row 3 = 987. Digits used: {3,2,9,6,5,8,9,8,7}. Digits 8 and 9 are repeated. This combination is invalid.

Based on this exhaustive evaluation, there are exactly two combinations that satisfy all the given conditions:

1. Row 1 = 273, Row 2 = 546, Row 3 = 819
2. Row 1 = 327, Row 2 = 654, Row 3 = 981

Thus, 2 such combinations are possible.

Incorrect Options:

Options 1, 2, and 4 are incorrect because a systematic analysis of all potential first-row numbers, in conjunction with the conditions of numerical relationships between rows and the non-repetition of digits across all three rows, definitively yields only two valid combinations.