Consider the following three statements about offices and carpeting on the 9th floor of a building: I. All the offices on the 9th floor have wall-to-wall carpeting. II. No wall-to-wall carpeting is pink in colour. III. None of the offices on the 9th floor has pink wall-to-wall carpeting. If the first two statements are true, then the third statement is:

Introduction / Context:
This problem checks your understanding of basic logical inference with sets and subsets. It asks whether a third statement about offices and pink carpeting must be true once we accept two earlier statements as facts. Such questions train you to reason carefully instead of relying on intuition alone.

Given Data / Assumptions:

• Every office on the 9th floor has wall-to-wall carpeting (9th-floor offices are a subset of carpeted offices).

• No wall-to-wall carpeting is pink (pink carpeting does not exist anywhere in the building).

• The third statement claims that none of the 9th-floor offices has pink wall-to-wall carpeting.

• We must decide whether this third statement necessarily follows from the first two statements.

Concept / Approach:
Use simple set logic. Let “9th-floor offices” be one set, and “offices with wall-to-wall carpeting” be another. The first statement puts all 9th-floor offices inside the set of carpeted offices. The second statement says this carpeted set has no pink elements. From these, we must see what is implied about pink carpeting on the 9th floor.

Step-by-Step Solution:
Step 1: From Statement I, every 9th-floor office has wall-to-wall carpeting. So, 9th-floor offices ⊆ carpeted offices. Step 2: From Statement II, there is no wall-to-wall carpeting that is pink. Thus, within the set of carpeted areas, the colour pink does not appear at all. Step 3: Because all 9th-floor offices are carpeted, they are entirely contained within the set of carpeted places that have no pink. Step 4: Therefore, it is impossible for any 9th-floor office to have pink wall-to-wall carpeting. Step 5: This is exactly what Statement III asserts: none of the offices on the 9th floor has pink wall-to-wall carpeting.

Verification / Alternative check:
Imagine the building as a map. Colour every carpeted office gray, but do not allow any pink carpet anywhere. The 9th floor is entirely gray, with no exceptions. Since pink is banned from the carpeted set as a whole, it cannot suddenly appear only on the 9th floor. The third statement simply restates the combined effect of the first two statements focused on one floor.

Why Other Options Are Wrong:
Option B is wrong because the third statement is not always incorrect; it must be correct if the first two statements are true.
Option C is wrong because we can, in fact, determine the truth of the third statement directly from the first two; there is no ambiguity.
Option D is wrong because the third statement is not independent; it is tightly linked to the first two and follows logically from them.

Common Pitfalls:
Students sometimes think that because the third statement “sounds similar” to the first two, it might be redundant or impossible to classify. Others confuse “no wall-to-wall carpeting is pink” with “no pink exists anywhere in the building”, which is stronger than necessary. The key is to follow the subset relationships carefully instead of relying on vague impressions.

Final Answer:
Therefore, the third statement is a direct logical consequence of the first two and is always correct and must follow from the first two statements.