Final answer:
The first-line treatment option for a patient with pharyngitis, allergic to penicillin, is Azithromycin (Z-Pak) for 5 days.
Step-by-step explanation:
The first-line treatment for a patient with a positive rapid antigen detection test for pharyngitis, who is allergic to penicillin, would be Azithromycin (Z-Pak) for 5 days as option (c) suggests. Penicillin V and Amoxicillin should be avoided due to the patient's penicillin allergy. Though Cephalexin is another alternative for those allergic to penicillin, Azithromycin is preferred as it has a shorter course and is generally well tolerated. Additionally, Azithromycin is effective against Streptococcus pyogenes, the common causative agent for pharyngitis. Treatment should be taken as prescribed to avoid any risk of developing antibiotic-resistant bacteria.