We study the problem of selecting one of the r best of n rankable individuals arriving in random order, in which selection must be made with a stopping rule based only on the relative ranks of the successive arrivals. For each r up to r=25, we give the limiting (as n-->[infinity]) optimal risk (probability of not selecting one of the r best) and the limiting optimal proportion of individuals to let go by before being willing to stop. (The complete limiting form of the optimal stopping rule is presented for each r up to r=10, and for r=15, 20 and 25.) We show that, for large n and r, the optical risk is approximately (1-t*)r, where t*[approximate]0.2834 is obtained as the roof of a function which is the solution to a certain differential equation. The optimal stopping rule [tau]r,n lets approximately t*n arrivals go by and then stops 'almost immediately', in the sense that [tau]r,n/n-->t* in probability as n-->[infinity], r-->[infinity]
