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📅 February 26, 2026 👤 scraface

Feb 26, 2026

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📌 Thus, the virologist can form $\boxed{9}$ valid test panels.

📌 Question: A science fair judge ranks 5 student projects from best to worst, but two projects, A and B, are tied for rank 3. The judge randomly assigns the actual ranks under the constraint that no two projects are tied — meaning ties are broken, but in this case, the rule implies that the rank distribution must reflect the observed ties. However, the judge decides that exactly one tie occurs, and it must be between A and B. How many distinct ranking configurations (i.e., total orderings with ties) satisfy this?

📌 Solution: We are to count the number of distinct ranking configurations of 5 projects where:

📌 There is exactly one tie,

📌 The tie is between A and B,

📌 The other three projects have distinct ranks (superior or below the A-B pair).

📌 Since A and B are tied for rank 3, and all other projects have unique ranks, the full structure is:

📌 Two projects at rank 3 (A and B), then one project strictly above, two strictly below — or possibly three above, etc., but the key is A and B tie at rank 3, and no other ties.

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