In each of the following questions, there are three statements followed by…

Concept

Two syllogism rules apply here. A chain of universal ("All") statements sharing a middle term combines directly — All A are B and All B are C give All A are C. A particular affirmative ("Some") statement always converts — Some A are B gives Some B are A — but this conversion never carries over to a term nested inside one of the two original terms.

Application

1. Statements 1 and 2 are both "All" statements sharing the middle term "boxes": All bottles are boxes, All boxes are bags. Chaining them directly gives All bottles are bags — conclusion 3.

2. Statement 3, "Some bags are trays", is a particular affirmative statement, so it converts to "Some trays are bags" — conclusion 4.

3. Conclusion 1 ("Some bottles are trays") and conclusion 2 ("Some trays are boxes") both need the bags–trays overlap to specifically reach boxes or bottles. The statements never place that overlap inside boxes or bottles — it can sit entirely in the part of "bags" that is not "boxes" — so neither conclusion is forced.

Cross-check (Venn diagram)

Nest bottles inside boxes inside bags, then draw the trays circle overlapping only bags, clear of the boxes/bottles region, as in the valid layouts above. "All bottles are bags" and "Some trays are bags" hold in every such layout, while "Some bottles are trays" and "Some trays are boxes" can both be false — confirming only conclusions 3 and 4 are guaranteed.

Result

Only conclusions (3) and (4) follow, so "Only (3) and (4)" is the correct answer.