The action of a few permutations on r-tuples is quickly transitive

We prove that for every r and d ≥ 2 there is a C such that for most choices of d permutations π₁, π₂, . . . , π_d of S_n, the following holds: for any two r-tuples of distinct elements in {1, . . . , n}, there is a product of less than C log n of the π_is which map the first r-tuple to the second. Although we came across this problem while studying a rather unrelated cryptographic problem, it belongs to a general context of which random Cayley graph quotients of S_n are good expanders.