Scheduling Six PO Interviews Under Fixed Constraints
Six candidates - Rao (A), Iyer (B), Khan (C), Bose (D), Das (E) and Nair (F) - are to be interviewed for probationary officer vacancies on a single day, one per slot, in slots numbered 1 to 6. After complaints about the previous recruitment drive, the HR cell has fixed these conditions: (i) Rao must be interviewed immediately before Iyer, in consecutive slots; (ii) Khan must occupy an even-numbered slot; (iii) Bose must be interviewed before Khan; (iv) Das must not occupy a slot immediately adjacent to Rao's slot; (v) Nair must be interviewed in slot 6, because he arrives from another city by the afternoon bus. The chairperson wants a written schedule that can be defended if a candidate challenges the order, so every placement must be justified from the conditions rather than from convenience, and any request for a change must be answered with reasoning rather than by reshuffling the list.
1. Determine every schedule that satisfies all five conditions. Is any candidate's slot fixed regardless of the arrangement?
Read the worked answer
Yes - Das (E) must be in slot 5 in every valid schedule, and there are exactly three valid schedules.
Reasoning:
- Nair (F) holds slot 6, so the Rao–Iyer block (A,B) can only occupy (1,2), (2,3), (3,4) or (4,5).
- Khan (C) must be even and cannot use 6, so C ∈ {2, 4}.
Case C = 2: Bose (D) must precede C, so D = 1. The A–B block must then be (3,4) or (4,5). If (4,5), then E = 3, which is adjacent to A at 4 - barred by condition (iv). So A–B = (3,4), E = 5. → D C A B E F
Case C = 4:
- A–B = (1,2) → D = 3, E = 5 → A B D C E F
- A–B = (2,3) → D = 1, E = 5 → D A B C E F
- A–B = (5,6) is impossible because F holds 6.
All three schedules place E in slot 5. That is the defensible statement to put in writing: Das's slot is not a matter of preference, it follows from conditions (ii) to (v).
2. Das requests an early slot and asks to be placed in slot 2. Show whether this is possible, and draft the one-paragraph reply to him.
Read the worked answer
It is impossible, and the proof is short.
With E = 2 and F = 6:
- Khan (C) must be even, and 2 and 6 are taken, so C = 4.
- The A–B block needs two consecutive free slots from {1, 3, 5}. No two of those are adjacent, so no valid placement for the block exists.
Equivalently: C = 4 forces A–B into (1,2) or (2,3), and slot 2 is occupied by Das in both.
Reply to Das: 'We have re-examined the schedule against the five conditions fixed by the HR cell. Slot 2 cannot be assigned to you because Khan must take an even slot; with Nair fixed at slot 6, that places Khan at 4, and Rao and Iyer must then occupy either slots 1–2 or 2–3. Slot 2 is therefore structurally unavailable to any candidate other than Rao or Iyer. Your slot at 5 is not a preference but a consequence of the same rules, and it is the only slot that satisfies them. If the timing creates a genuine difficulty, we can discuss the reporting time at the venue, but the interview order itself cannot be varied.'
This answers the request with reasoning, which is what protects the bank if the order is challenged.
3. The chairperson wants Rao interviewed in slot 1 so she can join by video link. Is that consistent with the conditions? Give the schedule or the contradiction.
Read the worked answer
It is consistent - and it forces a unique schedule.
- A = 1 implies B = 2 (condition i).
- C must be even; 2 is taken and 6 is Nair's, so C = 4.
- D (Bose) must precede C, and slots 1 and 2 are taken, so D = 3.
- The only remaining slot is E = 5, and 5 is not adjacent to A at 1, so condition (iv) is satisfied.
Schedule: A B D C E F - Rao, Iyer, Bose, Khan, Das, Nair.
Two points worth minuting:
- The result is unique, so the chairperson should be told that accepting the video-link request removes all flexibility for everyone else; if Rao's link fails, the whole schedule moves and no alternative order exists with her in slot 1.
- This schedule happens to be one of the three already found, so no condition is being bent. That is the test to apply to every special request: not 'can we fit it in', but 'does a valid arrangement still exist, and is it unique'.