A solid piece of silver metal reacts with oxygen in the air and tarnishes. What type of reaction is it? 

Corrosion is generally a process in which refined metals on reaction with the air around them are converted into more stable compounds such as metal oxides, metal sulphides, or metal hydroxides. Similarly, rusting of iron involves the formation of iron oxides by exposure to atmospheric moisture and oxygen.

Now, silver articles also become black when exposed to air. This is because it reacts with Sulphur in the air to form a coating of silver sulphide. This phenomenon is the corrosion of silver.
The original color and lustre of silver metal are restored by rubbing the surface with toothpaste, as this removes the sulphide layer.
Therefore, Silver(Ag) reacts with hydrogen sulphide $\left({\mathrm{H}}_2 \mathrm{S}\right)$ present in the air to form a black-coloured layer of silver sulphide $\left({\mathrm{Ag}}_2 \mathrm{S}\right)$.

As a new substance is synthesized in this process, thus it is a synthesis reaction.

Hence, the correct option is (B).