Let $S_n$ be the symmetric group on $n$ elements. Now, we pick a random transpositions to generate random walks on $S_n$ (also assume the probability of picking each transposition is equal of course). There are $n(n-1)/2$ transpositions. How many does is need on average (expectation) to get back to the identity $(1)$? Namely, what is the expectation step of a random walk on $S_n$ first hit the identity?
I could compute the case for 2,3, and 4. The expectations are simply 2, 6, 24. So I guess it is $n!$.