If you think there is a chance there will be 6 of you, its probably a good idea to try and make a permit reservation. The camping zones are based on number of people, not number of groups. It would be the safe choice. The permit costs 25 bucks regardless of the amount of people. Also, if you want a popular campsite like Holly Lake (which only has 3 sites), then you definitely should reserve it.
Park Service does hold several spots open in each zone for walk up permits though. And the backcountry thins out after August. I would be shocked if you weren't able to get your permit the day of your trip that late in the year.
@Bob is right about camping outside the park without a permit if you plan it right. Several great places around Alaska Basin, Mount Meek, Fox Creek Pass, and Moose Lake. However, if you want to include Paintbrush Divide in your route (which I highly suggest) you would definitely need a campsite inside the park.