+54 In how many ways can three pairs of siblings from different families be seated in two rows of three chairs, if siblings may not sit next to each other in the same row, and no child may sit directly in front of their sibling?
+964 Let's consider the two rows separately.
Row 1:
There are 3! ways to arrange the 3 pairs of siblings.
Once the siblings are arranged, there are 2 ways to seat each pair (either the older sibling on the left or the younger sibling on the left).
So, there are 3! * 2^3 = 48 ways to seat the siblings in Row 1 without any siblings sitting next to each other.
Row 2:
Now, for each arrangement in Row 1, there are 2 ways to seat each pair in Row 2: either the sibling who is to the left of their sibling in Row 1 should sit on the left in Row 2, or they should sit on the right.
So, for each arrangement in Row 1, there are 2^3 = 8 ways to seat the siblings in Row 2 without any siblings sitting next to each other or directly in front of their sibling.
Therefore, the total number of ways to seat the three pairs of siblings is 48 * 8 = 384.
