The Greek mathematician Archimedes is most famous for discovering that the quantities of water displaced by an immersed object have the same volume as the object itself.

A. discovering that the quantities of water displaced by an immersed object have
B. his discovery that by immersing objects in water, they displace quantities of
C. discovering that the quantities of water displaced by an immersed object has
D. discovering the quantities of water displaced by an immersed object of
E. his discovery of the fact that an object displaces water quantities, when immersed, of

A - Well done!

In this answer choice the plural verb have agrees with the plural subject quantities.
B - Incorrect.

This answer choice is both illogical and stylistically flawed.

In the corrected sentence the original phrase an immersed object is replaced by the phrase by immersing objects in water. As a result, we expect the next part of the sentence to introduce the subject of the action immersing: someone who is immersing the objects in water. Instead, we encounter the pronoun they which can refer only to the plural noun objects. The result is illogical, because the objects can't immerse themselves.

In addition, the next part of the sentence is also illogical because it does not say quantities of what are displaced.

Furthermore, the pronoun his is an unnecessary and therefore redundant addition to the noun discovering.

C - Incorrect.

This answer choice is grammatically incorrect. The singular verb has does not agree with the plural subject quantities of water.

D - Incorrect.

This answer choice is illogical and it unnecessarily changes the meaning of the original sentence.

In the original sentence Archimedes discovers that the quantities of water have the same volume as the object, but according to the corrected sentence he simply discovers the quantities themselves, and not a general truth about what the quantities of water have in terms of volume.

In addition, the resulting phrase an immersed object of the same volume of the object itself does not make any sense here.

E - Incorrect.

This answer choice is illogical and stylistically flawed. The modifying phrase when immersed is a dangling modifier, because it is at a distance from the noun it describes, and therefore creates ambiguity. The phrase when immersed should come directly after the noun object., otherwise it may seem that water quantities are immersed, which makes little sense.