ORIGINAL: Don K
From the regulations page 13, left side half way down....................
PERMIT LIMIT
No hunter may receive, or attempt to receive, more than one either sex firearm permit and one antlerless-only permit prior to the random daily drawing in September. Permits remaining after the lotteries are issued by random daily drawings until the county permit quotas are exhausted.
So you "Can't" apply for multiple permits (counties etc.) until the regular first and second lotteries are complete.
You beat me too it, Don! He is correct. You can only can apply (and recieve) one either sex permit before the random daily drawings. Once the 2nd drawing is done you can apply for whatever counties you want. No matter how many permits you get you can only shoot two bucks total all year though. I had this question last year.