We study the natural action of $S_n$ on the set of $k$-subsets of the set $\{1,\dots, n\}$ when $1\leq k \leq \frac{n}{2}$. For this action we calculate the maximum size of a minimal base, the height and the maximum length of an irredundant base. Here a "base" is a set with trivial pointwise stabilizer, "height" is the maximum size of a subset with the property that its pointwise stabilizer is not equal to the pointwise stabilizer of any proper subset, and an "irredundant base" can be thought of as a chain of (pointwise) set-stabilizers for which all containments are proper.