Let k, n be natural numbers with k ≦ n/2 and let Xn,k denote the set of k-element subsets of {1, 2, … n}. The symmetric group Sn acts in a natural way on the set Xn,k. Motivated by a question of Robert Guralnick, we investigate the size of a minimal base for this action. We give constructions providing a minimal base if n = 2k or if n ≧ k2. We also describe a general process providing a base of size at most c times bigger than the size of a minimal base for some universal constant c