**Core Concept**
Hypocalcemia is a condition characterized by low calcium levels in the blood. Carpopedal spasm, a symptom of hypocalcemia, can be triggered by rapid shifts in calcium levels, such as those occurring after repeated episodes of vomiting. The parathyroid glands play a crucial role in regulating calcium levels by secreting parathyroid hormone (PTH), which stimulates the release of calcium from bones.

**Why the Correct Answer is Right**
The patient's symptoms are likely due to hypocalcemia caused by prolonged vomiting, leading to a shift of calcium from the extracellular space into cells. Administering calcium gluconate or calcium chloride can help replenish the extracellular calcium pool and alleviate symptoms. This is an appropriate treatment for acute hypocalcemia, especially in cases where the patient is at risk of developing tetany or carpopedal spasm.

**Why Each Wrong Option is Incorrect**
**Option A:** Administering magnesium sulfate would not directly address the patient's hypocalcemia, as it is primarily used to treat magnesium deficiency.
**Option B:** Phosphate binders are used to manage hyperphosphatemia, not hypocalcemia.
**Option C:** Activated charcoal may be used to treat poisoning, but it does not address the underlying cause of hypocalcemia.

**Clinical Pearl / High-Yield Fact**
In patients with hypocalcemia, the addition of glucose can help drive calcium back into the extracellular space by promoting the uptake of calcium into cells.

**Correct Answer: C. Calcium chloride.**