The correct answer is: A. no article, the
The Smiths enjoyed their rest at the coast of the Black Sea last summer.
The article “the” is used before a noun to indicate that the noun is specific or unique. In this case, the Smiths are not just visiting any coast, they are visiting the coast of the Black Sea. The Black Sea is a specific body of water, and there is only one Black Sea.
The article “a” is used before a noun to indicate that the noun is general or non-specific. In this case, the Smiths could be visiting any coast, not just the coast of the Black Sea.
There is no article before the noun “coast” because it is a general noun. The coast is not a specific place, it is a general area along the edge of a body of water.