**Core Concept**
Mucin is a high-molecular-weight glycoprotein that forms a protective gel-like layer on the surface of epithelial cells, particularly in the respiratory, gastrointestinal, and genitourinary tracts. The staining of mucin is crucial in the diagnosis of various conditions, including adenocarcinomas and other types of epithelial tumors.

**Why the Correct Answer is Right**
Mucin is stained with Alcian blue, a cationic dye that binds to the negatively charged sialic acid residues in mucin molecules. This binding is due to the electrostatic attraction between the positively charged dye and the negatively charged sialic acid groups. The resulting complex is then visible under the microscope, allowing for the identification of mucin-producing cells.

**Why Each Wrong Option is Incorrect**
**Option A:** Alizarin red is a dye used to stain calcium deposits, not mucin. It is commonly used in the diagnosis of conditions such as osteogenesis imperfecta and other bone disorders.
**Option B:** Periodic acid-Schiff (PAS) is a staining technique used to detect glycogen and other carbohydrates in tissues. While PAS can stain some types of mucin, it is not the most specific or sensitive stain for mucin.
**Option C:** Hematoxylin and eosin (H&E) is a general-purpose staining technique used to visualize the morphology of tissues. While H&E can provide some information about mucin-containing cells, it is not specific for mucin and can be misleading.

**Clinical Pearl / High-Yield Fact**
When staining mucin, it's essential to use a cationic dye like Alcian blue to take advantage of the electrostatic attraction between the dye and the negatively charged sialic acid residues in mucin molecules.

**Correct Answer:** C. Alcian blue