Which cocktail includes both Sweet and Dry Vermouth?

The Bronx Cocktail is the correct choice because it is specifically known for its combination of both sweet and dry vermouth. The cocktail typically includes gin, orange juice, sweet vermouth, and dry vermouth, which together create a balanced and flavorful drink. This dual use of vermouth enhances the complexity and depth of the cocktail, distinguishing it from others that may contain only one type of vermouth or none at all.

Brandy Alexander, for instance, traditionally includes brandy, crème de cacao, and cream, without any vermouth. The Brain Hemorrhage is a cream-based shot featuring peach schnapps and a cream liqueur, not related to vermouth at all. The Blue Hawaiian consists of rum, blue curaçao, pineapple juice, and sweet and sour mix, which similarly does not involve vermouth.

Thus, the Bronx Cocktail stands out as the only option that incorporates both sweet and dry vermouth, reinforcing its classic cocktail status.