How many ice cream cones?

Over at the Social Sciences Statistics Blog, Gary King proposes a set of problems in counting motivated by the possible ways of constructing orders in an ice cream shop. He quickly multiplies up to numbers that exceed the particle count for the entire universe.

Of course, the next step is to count the number of interesting choices, for example, triple-scoop ice cream cones, where the order of the flavors doesn’t matter, and the three flavors are all different. This should give a somewhat manageable number, quite a bit smaller than the number of particles in the universe.

For example, Baskin-Robbins 31 Flavors claims to have developed "over 1000 flavors" since they opened. This gives a lower bound for the number of differently-flavored triple scoops as {1001 choose 3} = 166,666,500. This is manageable for a store that’s open 24/7, serving one cone per second: they can do every combination in only 5.285 years.

Ah, but here’s the probability problem: If the customers choose their flavor combinations independently and at random, what’s the expected number of untasted combinations as a function of time?