Final answer:
The 4-year-old boy with hematomas, swollen knees, a family history of bleeding disorders likely has hemophilia, a genetic disorder characterized by a deficiency in clotting factors.
Step-by-step explanation:
Given the symptoms of hematomas, knee pain, swelling, and a family history of a bleeding disorder, the likely diagnosis for the 4-year-old boy is hemophilia. Hemophilia is a genetic disorder characterized by an inability to produce clots due to a deficiency in clotting factors. Hemophilia A, which is caused by a deficiency in factor VIII, accounts for approximately 80% of cases, while Hemophilia B is caused by a deficiency in factor IX and makes up about 20% of cases. Since the disorder is linked to the X chromosome, it is often passed from a carrier mother to her male offspring. The mother's knowledge of a family history of bleeding disorders suggests that her son might be affected by this condition.