Three statements are given followed by three conclusions (I), (II) and (III).…

Concept: A syllogism conclusion is valid only if it is forced by the premises, not merely plausible. Two rules apply here: (a) if one class sits wholly inside a second class, and a third class sits wholly outside that second class, the third class must also sit wholly outside the first class; (b) when two candidate conclusions are exact contradictories of each other (a 'Some X is Y' against a 'No X is Y' on the same pair of classes) and neither is individually forced by the premises, exactly one of them must still be true in reality, so 'Either the one or the other follows' becomes the valid combined conclusion.

1. 'All fruits are vegetables' places the fruit-class entirely inside the vegetable-class.

2. 'No apple is vegetable' places the apple-class entirely outside the vegetable-class.

3. Since fruits sit wholly inside vegetables and apples sit wholly outside vegetables, apples must also sit wholly outside fruits, so 'No apple is fruit' (conclusion II) is forced with certainty.

4. Conclusion I, 'Some bananas are fruits,' needs a direct link between the banana-class and the fruit-class. The only banana fact given is 'Some apples are bananas,' which shows bananas overlapping the apple-class, and apples are already proven disjoint from fruits. That overlap cannot certify anything about the fruit-status of bananas, so I is not forced.

5. Conclusion III, 'No fruit is banana,' needs the premises to rule out any banana-fruit overlap outright. Nothing given does that: the non-apple portion of the banana-class is left completely free to overlap fruits or not, so III is not forced either.

6. Notice that I ('Some banana is fruit') and III ('No banana is fruit') are exact contradictories on the same pair of classes. In reality exactly one of them must be true even though the premises alone cannot say which, which is precisely the either/or rule from the concept above.

Cross-check: Cross-check with a quick Venn picture: draw Vegetables as the outer boundary, Fruits entirely inside it, and Apple entirely outside it, so Apple and Fruit never touch, confirming (II). Now draw Banana overlapping Apple as required by the third statement; the rest of the Banana circle is unconstrained, so it may or may not dip into Fruit, which is exactly the two contradictory scenarios behind the (I)/(III) either-or.

Result: So conclusion (II) follows on its own, and between (I) and (III) exactly one must follow, matching the option that reads 'Only either conclusion (I) or (III), and (II) follow.'