Question:
Several senior officials spoke to the press on condition that they not be named in the story.
A. that they not be named
B. of not being named
C. they will not be named
D. that their names will not be used
E. that their names are not used

*** See below. Answer: A

Yep....
thanks... for the reply but the answer in C also contains they..... ?????
is C eliminated because it is meaningless???

The key is that because of the phrase "on condition that," the sentence requires the subjunctive. The subjunctive verb form is the bare form of the verb with no auxiliary and no -s, so in this case, it's "be." Not "will be" or "should be" or anything like that, just "be."

You use the subjunctive with verbs that express demands, wishes, etc., as in "I insist that he be there" or "It is essential that each person show up on time."

p.s. There are no pronoun problems in this sentence. Since "officials" is the only plural noun, it's OK to use "they" or "their." This question is just testing the use of the subjunctive.