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In how many ways can three pairs of siblings from different families be seated in two rows of three chairs, if siblings may sit next to each other in the same row, but no child may sit directly in front of their sibling?
To solve this problem, let's consider the different possibilities for seating arrangements:
First, let's think about the possible arrangements for one pair of siblings in one row of three chairs. The first child in the pair can sit in any of the three chairs, and then the second child has two remaining options. So there are a total of 3 * 2 = 6 arrangements for one pair of siblings in one row.
Now, let's consider the arrangements for two pairs of siblings in one row. We need to look at the different cases:
Case 1: Both pairs of siblings sit on the same side of the row. In this case, we can seat the first pair of siblings in one row using the 6 arrangements we found before. Then, the second pair of siblings can sit in the remaining two chairs of that row in 2 * 1 = 2 arrangements. So, there are a total of 6 * 2 = 12 arrangements in this case.
Case 2: The two pairs of siblings sit on opposite sides of the row. In this case, we can seat the first pair of siblings in one row using the 6 arrangements we found before. Then, the second pair of siblings can sit on the other row in 3 * 2 = 6 arrangements. So, there are a total of 6 * 6 = 36 arrangements in this case.
Adding up the arrangements from both cases, we have 12 + 36 = 48 arrangements for two pairs of siblings in one row.
Now, let's consider the arrangements for three pairs of siblings in two rows. Again, we need to look at the different cases:
Case 1: All three pairs of siblings sit on the same side of the rows. In this case, we can seat the first pair of siblings in one row using the 6 arrangements we found before. Then, the second pair of siblings can sit in the remaining two chairs of that row in 2 arrangements, and the third pair can sit in the remaining two chairs of the other row in 2 arrangements. So, there are a total of 6 * 2 * 2 = 24 arrangements in this case.
Case 2: Two pairs of siblings sit on one side of the rows, and the third pair of siblings sits on the other side. In this case, we can seat the first pair of siblings in one row using the 6 arrangements we found before. Then, the second pair of siblings can sit on the other row in 3 * 2 = 6 arrangements, and the third pair can sit on the same row as the second pair in 2 arrangements. So, there are a total of 6 * 6 * 2 = 72 arrangements in this case.
Adding up the arrangements from both cases, we have 24 + 72 = 96 arrangements for three pairs of siblings in two rows.
Therefore, there are a total of 96 possible ways to seat three pairs of siblings from different families in two rows of three chairs, given the given conditions.