Four-Person Squad Selection with Overlapping Rules
Seven candidates - A, B, C, D, E, F and G - are being considered for a four-person squad. Exactly four must be selected. The rules are: if A is selected, B must be selected; C and D cannot both be selected; at least one of E and F must be selected; if G is selected, D must be selected; B and F cannot both be selected; and if C is selected, G cannot be selected. No other restrictions apply. A squad leader proposes the team A, B, D, G and claims it satisfies every rule because A has B and G has D. Another candidate wants to know whether C and G can ever appear together. The selection board also asks how many valid teams contain A but not F, and what they are. The candidates must reason from the stated rules only, not from personal preference or seniority.
1. Can a valid four-person team contain both C and G? Explain.
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No. The rule if C is selected, G cannot be selected directly forbids C and G together. There is also a second conflict: if G is selected, D must be selected, but if C is selected, C and D cannot both be selected. So selecting G forces D, which then blocks C. Both lines of reasoning lead to the same conclusion: no valid team contains both C and G.
2. If A is selected and F is not, list all valid teams and explain why no others work.
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A selected forces B. F is excluded, and at least one of E and F must be selected, so E must be selected. That gives A, B, E plus exactly one more member from C, D or G, because the team must have exactly four.
- Add C: A, B, E, C is valid. C and D are not together, G is absent, and B and F are not together.
- Add D: A, B, E, D is valid. C is absent and G is absent.
- Add G: G forces D, so the team would need A, B, E, G, D - five people, not four. It also fails the exact-size rule.
So there are exactly two valid teams: A, B, E, C and A, B, E, D.
3. Assess the proposed team A, B, D, G. What single substitution could make it valid while keeping D and G?
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A, B, D, G is invalid because neither E nor F is selected, so the rule at least one of E and F is broken. Keeping D and G means G’s requirement is satisfied, but the team still needs E or F and must remain size four. If A is replaced by E, the team becomes B, D, E, G. This is valid: E satisfies the E-or-F rule, C is absent, G has D, and B and F are not together. Replacing A by F fails because B and F cannot both be selected and A’s replacement does not fix A’s own rule cleanly; replacing B by E fails because A would then lack B. The clean single substitution is A out, E in.