Final answer:
The condition described as a solitary painful lytic bone lesion with overlying swelling and hypercalcemia in a child is Langerhans cell histiocytosis (Eosinophilic granuloma).
Step-by-step explanation:
The question asks about a solitary painful lytic bone lesion with overlying swelling and hypercalcemia in a child, and poses multiple-choice options to identify the condition. Among the choices listed, the correct answer would be d) Langerhans cell histiocytosis (Eosinophilic granuloma).
This condition, part of a group of disorders called Langerhans cell histiocytosis, typically presents with lytic bone lesions, which can be solitary and painful, often with associated swelling in the affected area. Hypercalcemia is a known complication due to increased osteoclast activity provoked by infiltrating Langerhans cells.