Given a friendship group of $n$ people, the binomial coefficient ${n \choose k}$ can tell me how many combinations of $k$ friends there are who could meet, e.g., if there are $10$ people and we meet in groups of $6$, that's $${ 10 \choose 6} = \frac{10!}{6!(10-6)!} = \frac{10 \times 9 \times 8 \times 7}{24} = 10 \times 3 \times 7= 210 $$ combinations.
However, what if $n$ people are actually $n/2$ couples and couples always remain together, e.g. at a dinner party. The actual answer now is simply a combination of $k/2$ couples from $n/2$ couples.
But if you want to explicitly retain the information $k/2$ distinct pairs, what might that look like in formal notation? It's a combination of subsets from a group of subsets but what would it look like?