I think the info that they are triplets/twins is somewhat strange, since there is a possibility that they are so-called identical triplets/twins, and thus could be considered in some sense non-distinguishable for counting purposes.

Nevertheless the given answer indicates that we do in fact view the siblings as distinct people, so:

Treat each set of siblings as a single unit, there are three such units, giving 3!

For the triplet, there are 3! ways to arrange them among themselves

For each pair of twins, there are 2! ways to arrange them among themselves

Thus overall we get 3! * 3! * 2! * 2! = 144.