They issue literally 1,000 or more permits each year and that doesn't put a dent in the population. If you are intererested you can go to
www.myfwc.com and look under alligators on the home page. The link will lead you to a sheet showing permits allowed for each individual body of water AND by county (additional permits).
In addition to gators we now have a resident breeding population of Burmese pythons here in Fla. This is from FWC's site today:
Armed with snake hooks and nets, a group of reptile experts selected by the Florida Fish and Wildlife Conservation Commission (FWC) to participate in the state's python permit program captured a 9-foot, 8-inch Burmese python. The volunteer permit holders spotted the python in water underneath a boardwalk leading to a camp on a tree island. It was later euthanized.