The first blank does not require an article because "Mary" is a proper noun that refers to a specific person who is already known. Proper nouns do not require an article.
The second blank requires the indefinite article "a" because "beautiful silver chain" is a singular countable noun, and we are referring to one chain in general.
The sentence becomes:
Mary was wearing a beautiful silver chain around her neck.
The correct answer is "No Article ... a".