To solve the problem, we start by labeling the siblings from the three different families as \( A_1, A_2 \) from family A, \( B_1, B_2 \) from family B, and \( C_1, C_2 \) from family C. Our goal is to arrange these six children in two rows of three chairs while adhering to the seating restrictions.
Step 1: Count the total number of arrangements without restrictions.
If we simply want to arrange six distinct children, the total number of arrangements would be: \[ 6! = 720. \]
Step 2: Apply the restrictions.
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Restricting siblings from sitting in the same row: Since we have 3 pairs of siblings, if any pair is seated in the same row, there are 2 ways (one on each side) to do this. Thus, we want to ensure each pair is seated in different rows.
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Valid row arrangements: We can specify the rows as Row 1 and Row 2. Each row must contain one child from each family.
We can choose one child from each family and the selection for Row 1 and Row 2 can be done in a specific order:
- Choose one from family A, one from family B, and one from family C for Row 1: The number of ways to choose one sibling from each family is \( 2 \times 2 \times 2 = 8 \).
- The remaining siblings must sit in Row 2, which also works out to be one from each family, and will automatically be the sibling not selected for Row 1.
Thus, the arrangement of choosing the siblings ensures that there are exactly 8 valid pairings of selections.
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Seating the selected siblings: Next, we must position the selected siblings in their respective rows. Each row has 3 chairs and can be filled in \( 3! \) (factorial of 3) ways.
Therefore, the arrangements per row will multiply as follows: \[ 3! \times 3! = 6 \times 6 = 36. \]
Step 3: Combine the calculations.
Finally, we multiply the number of valid selections and arrangements: \[ 8 \text{ (choices of siblings)} \times 36 \text{ (ways to arrange them)} = 288. \]
Thus, the total number of ways to seat the three pairs of siblings in the specified configuration is: \[ \boxed{288}. \]