Final answer:
In the scenario provided, the third-degree burn would typically be the principal diagnosis over the second-degree burn due to its severity, despite its smaller total body surface area. However, the presence of respiratory failure, which requires intubation and ICU care, may alter this if we consider all injuries.
Step-by-step explanation:
The appropriate principal diagnosis for a patient who has been admitted with multiple injuries including burns, fracture, and respiratory failure due to smoke inhalation is determined by the circumstances of the admission. According to the guidelines for burns, the principal diagnosis is often the condition that requires the most resources.
In this case, third-degree burns are usually considered more severe and life-threatening due to the complete destruction of the epidermis and dermis, often requiring more complex treatment, such as grafting. Since the patient has both second and third-degree burns but is intubated and placed in the ICU for mechanical ventilation, the respiratory failure seems to be the most life-threatening and resource-intensive condition at the moment. However, in the context of coding and billing, the specific circumstances of each case might lead to different principal diagnoses. If we are focusing on the burns alone and not considering the respiratory failure, the third-degree burn, even at a smaller total body surface area, would typically be the principal diagnosis over a larger second-degree burn since it is the most serious.