Rib Fracture: This is the most likely diagnosis given the history of a fall, as rib fractures are a common consequence of blunt trauma to the chest. The pain is typically sharp and worsens with deep breathing or coughing.

Muscle Strain: The muscles between the ribs (intercostal muscles) can be strained during a fall, leading to pain. This diagnosis is likely if the pain is more diffuse and not localized to a specific rib.
Costochondritis: Inflammation of the cartilage that connects the ribs to the breastbone (sternum) can cause pain, especially if the fall resulted in direct trauma to this area.
Pneumothorax: Although less common than rib fractures, a pneumothorax (collapsed lung) can occur if a fractured rib punctures the lung. Symptoms include sudden chest pain and shortness of breath.

Hemothorax: Accumulation of blood in the pleural space can occur due to severe trauma and can be life-threatening if not promptly treated. Symptoms include chest pain, shortness of breath, and decreased blood pressure.
Pulmonary Contusion: A bruise of the lung tissue can occur after a severe chest injury and may lead to respiratory failure if not recognized and managed properly.
Aortic Injury: Although rare, a severe fall can cause a traumatic aortic dissection or rupture, which is immediately life-threatening and requires emergency surgical intervention.

Osteomyelitis: Infection of the bone can occur after a fracture, including rib fractures, especially if the skin is broken. This would typically present with additional symptoms such as fever and redness around the fracture site.
Tumor: Metastatic disease to the ribs or primary bone tumors can cause pain but would be an unusual first presentation after a fall unless there was a pre-existing condition.
Slipping Rib Syndrome: A condition where the ribs slip out of place, causing pain. It's less common and might be considered if other diagnoses are ruled out and the pain persists.