**Core Concept**
Coumarin-based rodenticides, such as warfarin, are vitamin K antagonists that inhibit the synthesis of vitamin K-dependent clotting factors in the liver. This leads to a decrease in the production of these factors, resulting in a prolonged prothrombin time (PT) and increased risk of bleeding.

**Why the Correct Answer is Right**
The boy's bloody stool is likely due to the consumption of the coumarin-based rodenticide, which has caused a decrease in the production of clotting factors, leading to mucosal bleeding. Vitamin K is essential for the gamma-carboxylation of these clotting factors, which is necessary for their proper function. In the absence of vitamin K, these factors are not able to bind calcium ions and participate in the coagulation cascade, resulting in a prolonged PT and increased risk of bleeding.

**Why Each Wrong Option is Incorrect**
* **Option A:** This option is not relevant to the scenario. The boy's symptoms are not due to a gastrointestinal issue, but rather a coagulopathy caused by the ingestion of a rodenticide.
* **Option B:** This option is incorrect because the boy's symptoms are not due to a deficiency in vitamin K. In fact, vitamin K is being administered to help reverse the effects of the rodenticide.
* **Option C:** This option is incorrect because the boy's symptoms are not due to a liver disease. The liver is responsible for producing clotting factors, but in this case, the problem is not with the liver, but rather with the vitamin K-dependent synthesis of these factors.

**Clinical Pearl / High-Yield Fact**
It's essential to remember that coumarin-based rodenticides can cause a prolonged PT and increased risk of bleeding due to their inhibition of vitamin K-dependent clotting factor synthesis. This can be reversed with fresh frozen plasma and vitamin K administration.

**Correct Answer: D.**