Environmental Grant Panel Selection
The city's environmental grant panel has a budget of 240,000 CZK and must fund exactly three of six projects. The projects are: A river cleanup (90,000 CZK), B school garden (60,000 CZK), C bike-repair workshop (70,000 CZK), D air-quality sensors (100,000 CZK), E composting programme (80,000 CZK), and F tree inventory (50,000 CZK). The panel's statutes impose three rules: (1) if A is funded, B must be funded because they share a coordinator; (2) C and D cannot both be funded because they would use the same supplier and exceed its capacity; (3) at least one of E or F must be funded to meet the equity requirement. The panel has received a motion to fund A, C, and E. A member asks whether there is any valid combination that includes both C and E.
1. Test the motion A, C, E against the rules. What is the result?
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Check the motion A, C, E. Costs: 90,000 + 70,000 + 80,000 = 240,000, so the budget is exactly met and exactly three projects are chosen. Rule 2 is satisfied because D is not funded. Rule 3 is satisfied because E is funded. Rule 1 fails: if A is funded, B must be funded, but B is absent. Therefore the motion is invalid. The budget fit does not cure the rule violation. A valid replacement would need B in place of C or E, but then the total and the equity rule must be rechecked - for example, A, B, E costs 230,000 and satisfies all three rules.
2. If A is funded, which projects must or cannot be funded? Explain.
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If A is funded, rule 1 forces B. That uses 90,000 + 60,000 = 150,000, leaving 90,000 and exactly one more project because the panel must fund exactly three. The third project must satisfy rule 3, so it must be E or F; C alone or D alone would leave no E or F. D is also impossible on budget: A + B + D = 250,000 > 240,000. C is impossible because A + B + C = 220,000 but rule 3 fails. Therefore the only valid sets containing A are A, B, E (230,000) and A, B, F (200,000). In both, C and D are excluded. This shows how a conditional rule plus an exact-size requirement can eliminate options that look affordable.
3. Is there a valid combination containing both C and E? If yes, give one and explain why it satisfies all rules.
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Yes. C + E = 70,000 + 80,000 = 150,000, leaving 90,000 for exactly one more project. D is excluded by rule 2 because C is funded. A is excluded because A would require B and then the panel would have four projects (A, B, C, E), violating 'exactly three'. B is allowed: C, E, B = 210,000, three projects, no A, no D, and E satisfies rule 3. F is also allowed: C, E, F = 200,000, three projects, no A, no D, and both E and F satisfy rule 3. So C, E, B and C, E, F are valid. The example shows that the C–D conflict does not rule out C altogether; it only rules out the pair.