Desk Allocation at the KCC Camp
The branch manager of a semi-urban branch must assign six officers - Arora, Bhandari, Chauhan, Desai, Esha and Farooq - to six verification desks numbered 1 to 6 in a single row for a two-day Kisan Credit Card camp. Desk 1 is nearest the entrance; desk 6 is nearest the strong room. An audit note fixes five conditions. Chauhan cannot be at desk 1 or desk 6 because she handles the cash-counting tray. Desai must sit immediately to the right of Farooq so the two can pass files hand to hand. There must be exactly two desks between Esha and Chauhan, to keep the disputed land-document files apart. Arora must occupy an end desk because he is the camp coordinator. Bhandari must be to the right of Esha but not at desk 6, since his shift ends early and the door near desk 6 must stay monitored until closing. The manager has already confirmed that exactly one arrangement satisfies every condition.
1. Determine the desk number of each officer. State which single condition does the most work in forcing a unique arrangement, and why.
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Step 1 - use the Esha–Chauhan condition first. 'Exactly two desks between them' means their desk numbers differ by 3. Chauhan cannot be at desk 1 or 6, so the possible (Esha, Chauhan) pairs are: (5,2), (1,4) and (2,5). Pair (6,3) fails because Bhandari must be to the right of Esha and no desk lies right of 6; pair (3,6) fails because Chauhan cannot sit at desk 6.
Step 2 - test each pair.
- Chauhan at 2, Esha at 5: Bhandari must be right of 5, so Bhandari = 6.
- Chauhan at 5, Esha at 2: desks 1, 3, 4 and 6 remain; Farooq–Desai must be adjacent, which forces Farooq = 3 and Desai = 4, leaving Arora and Bhandari for desks 1 and 6. Bhandari must be right of Esha, so Bhandari = 6.
- Chauhan at 4, Esha at 1: seats 2, 3, 5, 6 remain for Arora, Bhandari, Farooq, Desai. Arora must be at an end desk, and desk 1 is taken, so Arora = 6. Farooq–Desai must be adjacent, which leaves only Farooq = 2, Desai = 3. Bhandari takes desk 5.
Step 3 - apply the last condition. Bhandari cannot be at desk 6, which eliminates the first two branches. Only the third survives.
Answer: Desk 1 Esha, desk 2 Farooq, desk 3 Desai, desk 4 Chauhan, desk 5 Bhandari, desk 6 Arora.
The condition 'Bhandari is not at desk 6' does the most work: each of the three branches independently pushed Bhandari to desk 6 except one, so this single negative condition cuts three candidate arrangements down to one.
2. The manager swaps Arora and Bhandari. Which of the five stated conditions are now violated, and which still hold?
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Start from the confirmed arrangement: 1 Esha, 2 Farooq, 3 Desai, 4 Chauhan, 5 Bhandari, 6 Arora. After the swap, Arora moves to desk 5 and Bhandari to desk 6.
Check each condition against the new line-up (Esha 1, Farooq 2, Desai 3, Chauhan 4, Bhandari 6, Arora 5):
- Chauhan not at desk 1 or 6: Chauhan is at desk 4 - still holds.
- Desai immediately to the right of Farooq: 3 is immediately right of 2 - still holds.
- Exactly two desks between Esha and Chauhan: desks 2 and 3 lie between desks 1 and 4 - still holds.
- Arora at an end desk: Arora is now at desk 5, which is not an end desk - violated.
- Bhandari right of Esha but not at desk 6: Bhandari at 6 is to the right of Esha at 1, but he is now at desk 6 - partly violated.
Answer: two conditions fail - Arora is no longer at an end desk, and Bhandari now occupies desk 6. The other three survive untouched. The trap is to assume that any swap disturbs the Farooq–Desai adjacency; it does not, because those two officers were not moved.
3. If the condition 'Bhandari is not at desk 6' is withdrawn and all other conditions stand, how many valid arrangements exist? List them.
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Re-run the three branches identified earlier without the final filter.
Branch A (Chauhan 2, Esha 5): Bhandari must be right of Esha, so Bhandari = 6. Arora must take an end desk; desk 6 is taken, so Arora = 1. Farooq and Desai fill 3 and 4 in that order. Arrangement: 1 Arora, 2 Chauhan, 3 Farooq, 4 Desai, 5 Esha, 6 Bhandari.
Branch B (Chauhan 5, Esha 2): Farooq–Desai must be adjacent, and the only adjacent free pair is 3–4, so Farooq = 3, Desai = 4. Desks 1 and 6 remain for Arora and Bhandari; Arora must be at an end desk, either works, but Bhandari must be right of Esha, so Bhandari = 6 and Arora = 1. Arrangement: 1 Arora, 2 Esha, 3 Farooq, 4 Desai, 5 Chauhan, 6 Bhandari.
Branch C (Chauhan 4, Esha 1): as before, Arora = 6, Farooq = 2, Desai = 3, Bhandari = 5. Arrangement: 1 Esha, 2 Farooq, 3 Desai, 4 Chauhan, 5 Bhandari, 6 Arora.
Answer: three arrangements. Note that two of them place Bhandari at desk 6, which is exactly why the withdrawn condition was decisive - it is a filter, not a structural constraint, and removing it multiplies the solutions rather than shifting them.