Abstract: A $k$-set of a permutation $\pi\in S_n$ is a subset $I\in [n]$ of size $k$ which is itself permuted by $\pi$. Equivalently, $I$ is a product of a subset of the cycles of $\pi$. In this talk, we discuss two problems: (1) If one chooses $r>1$ permutations at random, what is the likelihood that for some large $k$ each contains a $k$-set? This has application to the problem of invariable generation of $S_n$, and is connected with a famous old problem of Erdős: to show that almost all integers have two divisors in some dyadic interval $(y,2y]$. (2) Given $k_1, k_2, \ldots, k_m$ what is the likelihood that a random $\pi$ has a $k_1$-set, $k_2$-set, ..., $k_m$-set (all disjoint)? Such bounds are applied to the problem of estimating how many permutations $\pi \in S_n$ lie in transitive subgroups of $S_n$ other than $S_n$ or $A_n$. This is joint work with Sean Eberhard, Ben Green and Dimitrios Koukoulopoulos.