In the given question, a statement is followed by two conclusions. Decide…

Concept: In syllogisms, a conclusion follows only if it necessarily holds in every diagram consistent with the given statements. For two conclusions about the same subject and predicate -- one in the affirmative-particular form ("Some S are P") and one in the negative-universal form ("No S is P") -- the two are logical contradictories: in any consistent diagram exactly one of them must be true. When neither conclusion holds on its own but they form such a contradictory pair, the rule is that "Either conclusion follows".

Application: "Some apples are lichi" establishes only that apples and lichi overlap -- at least one apple is a lichi -- not that lichi is contained within apples. "No lichi is sweet" then removes that entire lichi group from sweet. Combining the two only fixes that this overlapping part (the apples that are lichi) is not sweet (giving the definite but unlisted conclusion "Some apples are not sweet"); it says nothing about the apples that are not lichi. Conclusion I ("Some apples are sweet") and Conclusion II ("No apple is sweet") are exactly this affirmative-particular/negative-universal contradictory pair about apples and sweet, so -- even though neither is individually guaranteed -- one of the two must hold in every diagram consistent with the statements.

• Both conclusions follow: fails -- a "some" statement and its "no" contradictory about the same subject-predicate can never both be true at once, so no valid diagram supports both together.

• Only conclusion I follows: fails on its own -- a diagram where none of the non-lichi apples are sweet (so no apple at all is sweet) is equally consistent with both statements, defeating this as the sole outcome.

• Only conclusion II follows: fails on its own -- a diagram where some non-lichi apples are sweet is equally consistent with both statements, defeating this as the sole outcome.

• Either conclusion I or II follows: holds -- since the two conclusions are contradictories on the same subject-predicate, exactly one of them is true in every diagram consistent with the statements, which is precisely the either-or condition.