Project ForgeFive persons P, Q, R, S and T are sitting in a row not necessarily in the same order. Q and R are separated by one person, and S should not be seated adjacent to Q. The number of distinct seating arrangements possible is:
- A4
- B8
- C10
- 16
Explanation
Q and R must occupy positions exactly 2 apart: the valid unordered position-pairs among 5 seats are {1,3},{2,4},{3,5}, each with 2 orderings of Q/R = 6 placements total. For each placement, S must avoid any seat adjacent to Q's actual seat, and the remaining people (P,S,T) fill the leftover 3 seats. Working through all 6 cases (accounting for how many seats are 'adjacent to Q' and available, which varies by whether Q is at an end or middle position) and multiplying by the 2 ways to arrange the remaining two people (P,T) once S's seat is fixed, gives 4+2+2+2+2+4 = 16 total valid arrangements.
Source: GATE ME 2021 Session 1 (ME-1) (2021)
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