Labelled and Unlabelled In combinatorics, what is the difference between labelled and unlabelled families? Here on question five, it found that there are 256 labelled families of subsets of a $3-$set and $20$ unlabelled families. I have no idea what labelled and unlabelled are. The book doesn't give a formal definition, instead it gives examples which aren't clear to me.

A labeled finite set $S$ is a set together with a bijective function $S\to [|S|]$. Deciding when two of these sets are the same depends on the context. For example, we would say that two labeled graphs are the same if there is an isomorphism between them preserving the labeling of the vertices (the bijective function mentioned above). In the context of your question it seems that the labeling on the family arises from a labeling of the whole set, so the labeling of the whole set would restrict to a function on the subsets in the family. In this case two labeled families are the same if they contain the same sets. For the unlabeled version of the problem, two families would be considered the same if there is a bijection from the whole set to itself inducing a bijections between the sets in the two families.