We can divide this problem into two cases:
Case 1: Each row has exactly one child from each family.
Choose one child from each family for the first row: there are 3 choices
.
Arrange the remaining two children in the first row (siblings can't be together): there are 2!=2 ways.
Arrange the second row similarly: 3⋅2=6 ways total.
Case 2: One row has two children from the same family.
There are two subcases depending on the arrangement of siblings in the rows:
Subcase 2a: The first child in each row is from the same family.
Choose one pair of siblings: 3 choices
.
Arrange the siblings within the pair: 2 ways.
Arrange the remaining 4 children (2 from another pair and 2 from the third pair) in the second row: 4! ways.
Overcount: we've counted the arrangement as if sibling order matters within a pair (which it doesn't) twice (once for each sibling in the first row). So, we divide by 2!⋅2!.
Total: 2!⋅2!3⋅2⋅4!=36 ways.
Subcase 2b: The first child in the first row is NOT a sibling of the first child in the second row.
Choose one pair to have their children occupy the third seats in each row: 3 choices.
Arrange the remaining 4 children in the first row: 4! ways.
Overcount: similar to subcase 2a, we divide by 2!⋅2! to account for sibling order not mattering.
Total: 2!⋅2!3⋅4!=36 ways.
Since Cases 1 and 2 are mutually exclusive, to find the total number of arrangements, we simply add the number of arrangements from each case:
6 + 36 + 36 = 78.
This gives us a final answer of 78.