**Core Concept**
FNAC (Fine-Needle Aspiration Cytology) is a diagnostic tool used to evaluate thyroid nodules for malignancy. However, its accuracy can vary depending on the type of thyroid carcinoma present.

**Why the Correct Answer is Right**
The least diagnostic thyroid carcinoma type by FNAC is the **medullary thyroid carcinoma (MTC)**. This is because MTC cells do not produce thyroglobulin, a protein that is often used as a marker in FNAC samples. Additionally, MTC cells can exhibit a variety of morphological patterns, making them more challenging to diagnose using FNAC alone. The diagnostic challenge lies in the fact that MTC cells can be easily mistaken for normal or reactive follicular cells, leading to false-negative results.

**Why Each Wrong Option is Incorrect**
* **Option A:** Papillary thyroid carcinoma (PTC) is generally well-diagnosed using FNAC due to the presence of distinctive nuclear features such as overlapping nuclei, intranuclear grooves, and pseudoinclusions.
* **Option B:** Follicular thyroid carcinoma (FTC) can be diagnosed using FNAC if there is evidence of capsular or vascular invasion, but this is not always possible to assess on a cytology sample.
* **Option D:** Anaplastic thyroid carcinoma (ATC) is often diagnosed clinically due to its aggressive presentation and rapid growth, but FNAC can still be used to confirm the diagnosis.

**Clinical Pearl / High-Yield Fact**
It's essential to remember that FNAC results should always be interpreted in the context of clinical and radiological findings. A combination of imaging studies, clinical evaluation, and histopathological examination is often necessary to make an accurate diagnosis of thyroid carcinoma.

**Correct Answer:** C. Medullary thyroid carcinoma.