**Core Concept**
The principle of using mannitol to calculate extracellular fluid (ECF) volume is based on the concept of dilutional measurement, where the amount of mannitol distributed in the ECF is used to estimate the total volume of the ECF compartment. This technique relies on the fact that mannitol distributes evenly throughout the ECF and is not significantly metabolized or taken up by cells.

**Why the Correct Answer is Right**
The formula used to calculate ECF volume is: ECF volume (ml) = (Total amount of mannitol injected in mg) x (Total body weight in kg) / (Concentration of mannitol in mg/100 ml x (1 - Fraction excreted)). In this scenario, the total amount of mannitol injected is 10 gm (10,000 mg), and the concentration measured is 50 mg/100 ml. Assuming a body weight of 70 kg, the ECF volume can be calculated as: ECF volume (ml) = (10,000 mg) x (70 kg) / (50 mg/100 ml x (1 - 0.1)).

**Why Each Wrong Option is Incorrect**
* **Option A:** This option is incorrect because it does not account for the fraction of mannitol excreted in the calculation.
* **Option B:** This option is incorrect because it uses an incorrect formula for calculating ECF volume.
* **Option C:** This option is incorrect because it does not take into account the total body weight in the calculation.
* **Option D:** This option is incorrect because it does not accurately reflect the formula used to calculate ECF volume.

**Clinical Pearl / High-Yield Fact**
When using mannitol to estimate ECF volume, it's essential to remember that a significant fraction of mannitol may be excreted by the kidneys, which can affect the accuracy of the measurement. A minimum of 2 hours should be allowed for mannitol to equilibrate in the ECF before measuring its concentration.

**Correct Answer: C. 10,000 x 70 / (50 x 0.9) = 13888 ml**