Yep, what he said about alternate dates, fill in all of them that you are able to go on.
Having more people in your group can make it less likely for you to get a permit, because there is a set number of people allowed to launch on the river for the date you request, so for instance, if the allotment were 50 people launching april 20th, and the total number of people listed on permit applications drawn before yours equals, say 45, and you asked to take a group of 7 boating, then your out of luck.
Not entirely positive how exactly the drawing takes place, but my guess is that's it's randomly done by a computer.
Of course you can keep an eye on rec.gov after you don't get picked, and pick up a cancelation sometimes( once they release those). I never one the lottery, but have gotten a couple cancelation permits.
Hope this actually makes sense, and is helpful.
Oh, the other factor about how many people you plan to have on your trip, is if they are also putting in for permits, then it may actually improve your odds to have more friends.
" I wish I were a headlight, on a North bound train..."