**Core Concept**
The question tests the student's ability to accurately measure and administer a specific dose of ceftriaxone using a calibrated syringe. This involves understanding the concentration of the medication, the volume of the syringe, and the desired dose.

**Why the Correct Answer is Right**
To solve this, we first need to calculate the concentration of ceftriaxone in the syringe. The concentration is 500 mg/5 mL, which simplifies to 100 mg/mL. Since the patient requires 180 mg, we need to find out how many mL of the solution this corresponds to: 180 mg / 100 mg/mL = 1.8 mL. Given that the syringe is 2 mL and has 10 divisions per mL, we can calculate the number of divisions to fill by multiplying the milliliters by the divisions per milliliter: 1.8 mL * 10 divisions/mL = 18 divisions. Therefore, the correct answer is to fill 18 divisions in the 2 mL syringe.

**Why Each Wrong Option is Incorrect**
**Option A:** This option is incorrect because it does not accurately calculate the concentration of ceftriaxone in the syringe. The correct concentration is 100 mg/mL, not 50 mg/mL.
**Option B:** This option is incorrect because it does not take into account the correct concentration of ceftriaxone in the syringe. The correct concentration is 100 mg/mL, not 200 mg/mL.
**Option C:** This option is incorrect because it does not accurately calculate the number of divisions to fill. The correct calculation is 1.8 mL * 10 divisions/mL = 18 divisions, not 20 divisions.

**Clinical Pearl / High-Yield Fact**
When administering medications, it's essential to accurately measure the dose using calibrated equipment to prevent errors and ensure patient safety. This includes using the correct concentration and volume of the medication, as well as the correct number of divisions to fill in a syringe.

**Correct Answer:** C. 18