Five Finalists, One Rule Too Many
Five finalists - Aman, Bela, Chirag, Divya and Esha - must speak one after another in slots 1 to 5 at a college debate. The organising committee has circulated these rules: Bela speaks immediately before Chirag; Esha is neither first nor last; exactly two speakers separate Divya and Esha; Aman speaks before Divya; and Chirag does not speak last. A first-year volunteer works out one order and submits it. A senior member says the rule about Chirag is unnecessary and that the order can be fixed from the other four rules alone. The cultural secretary, who has to print the schedule within the hour, wants a written check rather than an opinion, and also wants to know how much freedom the committee would have if the rule 'Aman speaks before Divya' were dropped. No speaker may share a slot and no stated rule may be silently relaxed.
1. Determine the speaking order using all the rules, showing how each case is eliminated.
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'Exactly two speakers separate Divya and Esha' means their slot numbers differ by 3. Within 1–5 the only pairs are (1, 4) and (2, 5).
Case 1 - Divya 1, Esha 4 (Esha cannot be first or last, so this orientation is the only one for the pair 1 and 4). Remaining slots 2, 3, 5 must take Aman, Bela and Chirag, with Bela immediately before Chirag. The only consecutive pair available is (2, 3), so Bela is 2 and Chirag is 3, leaving Aman at 5. But Aman must speak before Divya, and 5 is after 1. Contradiction - eliminate.
Case 2 - Esha 2, Divya 5. Remaining slots 1, 3, 4 take Aman, Bela and Chirag; the consecutive pair must be (3, 4), giving Bela 3 and Chirag 4, so Aman is 1. Aman before Divya holds (1 before 5), and Chirag is not last.
Order: 1 Aman, 2 Esha, 3 Bela, 4 Chirag, 5 Divya.
2. Is the senior member right that 'Chirag does not speak last' is redundant? Prove your answer rather than asserting it.
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Yes. The rule is implied by the others and can be removed without adding a single new schedule.
Direct proof: suppose Chirag were in slot 5. Bela must be immediately before him, so Bela is in slot 4. Divya and Esha must then occupy two of slots 1, 2, 3 with a gap of exactly two speakers between them. No two numbers among 1, 2, 3 differ by 3, so the spacing rule cannot be met. Chirag in slot 5 is therefore impossible under the remaining rules.
Second check: the case analysis in the previous task never needed the rule. Case 1 collapsed on the Aman–Divya rule alone, and Case 2 already placed Chirag at 4. So the rule is doing no work.
For the printed schedule the practical point is that keeping a redundant rule is harmless but misleading - it suggests the committee deliberated over something that was never a live constraint.
3. If the rule 'Aman speaks before Divya' were dropped and the rest kept, how many valid schedules would exist? List them.
Read the worked answer
Two schedules.
Case 1 - Divya 1, Esha 4. The remaining slots 2, 3, 5 must hold Aman, Bela and Chirag with Bela immediately before Chirag, so Bela 2, Chirag 3, Aman 5. This order was rejected before only because Aman came after Divya; with that rule gone it is valid. Check the rest: Esha is neither first nor last, Divya and Esha are three slots apart, Bela precedes Chirag immediately, Chirag is not last.
Case 2 - Esha 2, Divya 5. As before: Aman 1, Bela 3, Chirag 4.
The two surviving orders are: Aman, Bela, Chirag, Esha, Divya and Aman, Esha, Bela, Chirag, Divya.
So the Aman–Divya rule is what makes the schedule unique. If the cultural secretary wants a single printed order rather than a choice, that is the rule to keep and to defend publicly.