Five persons P, Q, R, S and T are to be seated in a row, all facing the same direction, but not necessarily in the same order. P and T cannot be seated at either end of the row. S should not be seated adjacent to P. R is to be seated at the second position from the left end of the row. The number of distinct seating arrangements possible is: A. 2 B. 3 C. 4 D. 5

GATE 2021 · General Aptitude · Seating Arrangement · medium

Answer: 3 distinct seating arrangements (Option B): Q-R-P-T-S, S-R-P-T-Q, S-R-T-P-Q

1. Fix R at seat 2; determine forced placements: R is at seat 2. Remaining: {P, Q, S, T} in seats {1, 3, 4, 5}. P and T cannot be at ends (seats 1 or 5). The only non-end available seats are 3 and 4. So P and T must occupy seats 3 and 4 in some order: (P=3,T=4) or (P=4,T=3). Q and S then fill seats 1 and 5.
2. Check each candidate against S-not-adjacent-to-P: Candidate 1: P=3, T=4, Q=1, S=5. |pos(S)-pos(P)| = |5-3| = 2. S not adjacent to P. Valid. Candidate 2: P=3, T=4, Q=5, S=1. |pos(S)-pos(P)| = |1-3| = 2. S not adjacent to P. Valid. Candidate 3: P=4, T=3, Q=1, S=5. |pos(S)-pos(P)| = |5-4| = 1. S IS adjacent to P. Invalid. Candidate 4: P=4, T=3, Q=5, S=1. |pos(S)-pos(P)| = |1-4| = 3. S not adjacent to P. Valid.
3. Count valid arrangements: Valid arrangements: Candidate 1 (Q-R-P-T-S), Candidate 2 (S-R-P-T-Q), Candidate 4 (S-R-T-P-Q). Total = 3.