OK ... the exemption card was dependent upon the room you chose. Everyone automatically assumed that one of the non-snipers had the exemption card. Isn't it possible that the exemption card was in the room that ended up having the sniper?
I think it's fairly likely that Mark was given a plausible answer to the puzzle beforehand, thus ensuring that HE got the exemption, or at least that no one else had the shot at it. He also knew which room NOT to go in, so as to ensure that someone else got the exemption card.