1) Why would they be visiting Reno casinos if they are visiting Nellis AFB? Nellis is right down the road from Las Vegas, not Reno.
2) When both Cpt. Nelson and Cpt. Healey get promoted to Majors, Maj. Healey gets the proper rank insignia on his shoulder, the gold oak leaf. But Maj. Nelson is wearing the insignia of a Lieutenant Colonel, the silver oak leaf. I'm assuming the producers did this simply for looks, as the silver does seem to go better with the blue Air Force uniform of Nelson, and the gold with the green Army uniform of Healey. But that is silly, as I believe the military was more strict with the inaccuracies of Hollywood in the 60s than they are today.