Scheduling Five Seminars
Five students - Ayesha, Bilal, Chen, Danish and Emaan - must each give one seminar on a different weekday from Monday to Friday. The coordinator has fixed Chen on Wednesday because the visiting examiner is available only that day. The other rules are: Ayesha must speak before Chen; Bilal must speak immediately before Emaan; and Danish refuses Monday and Friday because of a part-time job. No day can hold two seminars, and the coordinator wants a single confirmed schedule before booking the hall.
1. Determine the complete schedule. Show how each rule forces a placement.
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Chen is on Wednesday. Bilal and Emaan must occupy two consecutive days. The only available consecutive pairs are Monday–Tuesday and Thursday–Friday, since Tuesday–Wednesday and Wednesday–Thursday are blocked by Chen.
If Bilal–Emaan were Monday–Tuesday, Ayesha and Danish would take Thursday and Friday. Danish cannot take Friday, so Danish would be Thursday and Ayesha Friday - but Ayesha must speak before Chen on Wednesday. Contradiction.
So Bilal is Thursday and Emaan Friday. Monday and Tuesday remain for Ayesha and Danish. Danish cannot take Monday, so Danish is Tuesday and Ayesha Monday.
Final order: Monday Ayesha, Tuesday Danish, Wednesday Chen, Thursday Bilal, Friday Emaan.
2. If the rule 'Ayesha speaks before Chen' were deleted, how many schedules would satisfy the remaining rules?
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Two schedules.
Chen remains on Wednesday. Bilal–Emaan must still be either Monday–Tuesday or Thursday–Friday.
Case 1: Bilal Monday, Emaan Tuesday. Then Ayesha and Danish fill Thursday and Friday. Danish cannot take Friday, so Danish is Thursday and Ayesha Friday. This is valid once the Ayesha–Chen rule is gone.
Case 2: Bilal Thursday, Emaan Friday. Then Ayesha and Danish fill Monday and Tuesday. Danish cannot take Monday, so Danish is Tuesday and Ayesha Monday.
Both satisfy every remaining rule, so the answer is exactly two.
3. Replace 'Danish does not speak on Monday or Friday' with 'Danish does not speak on Friday.' Does the schedule remain uniquely determined? Explain.
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No, uniqueness is lost.
Chen is still Wednesday. Bilal–Emaan must be Monday–Tuesday or Thursday–Friday. If Bilal–Emaan were Monday–Tuesday, Ayesha and Danish would occupy Thursday and Friday, so Ayesha could not speak before Chen. That case fails.
So Bilal is Thursday and Emaan Friday, leaving Monday and Tuesday for Ayesha and Danish. The weakened rule allows Danish on Monday (only Friday is forbidden), so there are two valid orders: Ayesha Monday and Danish Tuesday, or Danish Monday and Ayesha Tuesday. Both keep Ayesha before Chen. The original rule was doing real work: it eliminated one of these two orders.