I also don't discard old, valid pairs of socks when I buy a new batch, so I end up with K>>1 varieties of socks in the drawer. Still, for any drawer containing M pairs of socks, we typically find that K < M/2, and often much smaller than that.
Inasmuch as my socks are typically worn, washed, and stored in pairs, the whole question of pulling more than one matching sock in the morning never comes up. Besides that question has already been solved, so today I can simply use socks; and leave the actual counting as an exercise for the grad students. Matching those socks to my outfit in the dark, however, remains NP-hard.