Logical syllogism — determine which conclusions follow beyond doubt. Statements: 1) Some keys are staplers. 2) Some staplers are stickers. 3) All the stickers are pens. Conclusions: (1) Some pens are staplers. (2) Some stickers are keys. (3) No sticker is key. (4) Some staplers are keys.

Introduction / Context:
This syllogism combines two particular premises with one universal affirmative about stickers and pens. We need to identify definite conclusions and recognize a classic “either–or” case.

Given Data / Assumptions:

• Some keys ∩ staplers ≠ ∅.
• Some staplers ∩ stickers ≠ ∅.
• All stickers ⊆ pens.

Concept / Approach:
From “Some staplers are stickers” and “All stickers are pens,” the particular set of staplers that are stickers is also a subset of pens. Therefore, there exists at least one element that is both a pen and a stapler. Additionally, “Some keys are staplers” immediately yields “Some staplers are keys.” However, with respect to the relation between stickers and keys, the data permit either overlap or separation; thus an exclusive uncertainty arises: either some stickers are keys, or no sticker is a key, but we cannot fix which.

Step-by-Step Solution:
(4) “Some staplers are keys” — directly from Statement 1 by commutation; this must follow. (1) “Some pens are staplers” — since some staplers are stickers and all stickers are pens, those staplers are pens; hence some pens are staplers. (2) “Some stickers are keys” — not forced; could be true if the overlapping staplers are among the key-staplers. (3) “No sticker is key” — also not forced; could be true if the sticker-staplers are not among the key-staplers.

Verification / Alternative check:
Construct two models: Model A with an element that is key+stapler+sticker (then (2) true, (3) false); Model B where the staple-sticker element is different from the key-stapler element (then (3) true, (2) false). In both models, (1) and (4) are always true. Hence the definite outcome is (1) and (4), and either (2) or (3).

Why Other Options Are Wrong:
Options that pick only (1) and (2) or only (2) and (4) ignore the alternative possibility. The only option capturing the certainty plus the exclusive uncertainty is the “Only (1) and (4) and either (2) or (3)” choice.

Common Pitfalls:
Assuming transitivity of “Some …” overlaps through a middle term without checking whether the same elements are involved.

Final Answer:
Only (1) and (4) and either (2) or (3)