Five-Student Presentation Schedule
Five students-Ayesha, Bilal, Chen, Danish, and Ema-present one project each, Monday to Friday, one per day. The rules are: Ayesha presents before Chen; Bilal does not present on Monday or Friday; Danish presents immediately before Ema; Chen presents on Wednesday. The supervisor adds that whoever presents on Friday must have completed a lab-safety form. Only Bilal and Ema have completed it. The students want to know whether the Friday rule changes the only possible order, and whether the order survives a move of Chen to Thursday.
1. Determine the full presentation order and explain how the constraints force it.
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Chen is Wednesday. Danish and Ema must occupy adjacent days with Danish first. The pair cannot be Tuesday-Wednesday or Wednesday-Thursday because Chen occupies Wednesday, so the only possible pairs are Monday-Tuesday or Thursday-Friday. If the pair were Monday-Tuesday, then Ayesha would need a day before Chen but Monday and Tuesday would already be taken, which is impossible. Therefore Danish is Thursday and Ema is Friday. Ayesha must be before Chen on Wednesday, so Ayesha takes Monday and Bilal takes Tuesday, since Bilal cannot take Monday or Friday. The order is Ayesha, Bilal, Chen, Danish, Ema. Friday is Ema, who has the form, so the safety rule is satisfied.
2. Is the lab-safety rule necessary to fix the order? Justify.
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No, it is redundant given the other rules. The Danish-Ema adjacency and Chen's Wednesday placement already force Danish to Thursday and Ema to Friday, as shown by eliminating the Monday-Tuesday branch. Therefore Friday is Ema whether or not the safety rule exists, and Ema has completed the form. The safety rule does not eliminate any additional arrangement. It would matter only if the other constraints allowed a different Friday presenter, such as Bilal; under the stated rules, that possibility is already blocked.
3. Chen is moved to Thursday. Does a valid order exist? Show the failed branches.
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No valid order exists. With Chen on Thursday, Danish-Ema can be Monday-Tuesday or Tuesday-Wednesday. If they are Monday-Tuesday, the remaining days for Ayesha and Bilal are Wednesday and Friday; Ayesha must be before Chen, so Ayesha takes Wednesday and Bilal takes Friday, which is forbidden. If they are Tuesday-Wednesday, the remaining days are Monday and Friday; Ayesha can take Monday, but Bilal is again forced to Friday, which is forbidden. In both branches Bilal would satisfy the safety form, so the failure comes from the ban on Bilal presenting Friday, not from the form rule.