Bumping this because I've discovered something that needs correction:
I think this collects together most of what we've learned canonically about Denobulan family life:
...
Denobulan families are large and complex; the family of Phlox, a typical member of the culture, had 720 members, with 42 romantic partnerships.
Actually Phlox didn't say his family had 720
members -- he said there were 720 different
relationships that existed among the various spouses, "42 of which have romantic possibilities." For instance, among Phlox and his three wives, there'd be one relationship each between Phlox and each wife, plus relationships between wife 1 and wife 2, between wife 1 and wife 3, and between wife 2 and wife 3, so that's six relationships among four people (twelve if you define, say, "Phlox's wife Feezal" and "Feezal's husband Phlox" as two separate relationships). Then there'd be each wife's relationships with her other two husbands, then her relationships with the other wife's two husbands, then each husband's relationship with each other husband, etc. So the number of relationships -- the number of possible pairings of distinct members of the family -- is significantly larger than the actual number of members of the family.
Now, the tricky part is breaking down the actual numbers here, combinatorically speaking. How did the scriptwriters of "A Night in Sickbay" (or their technical advisors) arrive at the number of 720 different relationship combinations, 42 of which could be sexual? I'm guessing it's got to do with combinatorics. We're talking about the number of possible combinations of
n marital partners taken 2 at a time, so we want n!/2!(n-2)! = 720 (where n! equals n-factorial, i.e. 4! = 1x2x3x4). Now, Phlox plus his three wives plus their two other husbands each is ten people, but 10!/2!(10-2)! = 45. Or if we disregard order (as in my Phlox/Feezal parenthetical) and do it as permutations, it becomes 10!/(10-2)! = 90 relationships.
I can get 720 if it's 10!/(10-3)!, but then we'd have to define a relationship as being between three people, and that doesn't make sense.
Okay, so what if we extend it one more tier? Each of those six husbands besides Phlox has up to an additional two wives, bringing the total number of partners to 22. 22!/2!(22-2)! = 231 -- still not there. Each of those 12 extra wives in turn has two other husbands, for 46 partners; 46!/2!(46-2)! = 1035. Now it's too much.
Okay, so I can't get combinatorics to work. I'm starting to think Phlox just got his math wrong. Either that or he's defining "relationship" in a way that can include three or more people. Anyone else want to take a stab at the math?