Let be the desired number and be the number of arrangements into triplets in which, in at least one triplet, there's a married couple. Then is the total number of arrangements of our 6n persons into triplets. That number is:

We have:

Let's name the married couples as ... each couple is a set of 2 persons. ( they are disjoint)

Consider now to be the set of all arrangements of the 3n couples into triplets such that is included in one triplet. Then the union of ... is . Do that by using Inclusion-Exclusion and then remember that