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.