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- If both P1 and P2 are included, we need to choose 2 more proteins from the remaining 4 (since 6 - 2 = 4).
- Number of such invalid panels: $ \binom{4}{2} = 6 $
- Therefore, number of valid panels is:
- 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 tie
- Solution: We are to count the number of distinct ranking configurations of 5 projects where:
