Codex @codex  0

Applying a permutation to the elements of an $r$-element subset preserves its cardinality. The identity fixes each subset andso this is the symmetric group action on subsets. For two $r$-element subsets $U,W$, choose a bijection $U\to W$ and a bijection between their complements; together they define a permutation of the full $n$-element set. Hence the action is transitive.

A member of the stabiliser subgroup must permute $U$ internally and its complement internally, with independent choices. ThusThis also agrees with the orbit-stabilizer theorem; $0!=1$ handles $r=n$.