Initial Seating Arrangement
Six people (P, Q, R, S, T, U) are around a circular table. We determine their initial positions based on the given clues.
- R is two places to the right of Q: If we place Q at a position, R is two steps clockwise from Q.
- P is three places to the left of R: P is three steps counter-clockwise from R. If Q is at position 1, R is at 3. Three steps left from R(3) lands P at position 6 (3 → 2 → 1 → 6).
- Current partial arrangement (clockwise): Q at pos 1, R at pos 3, P at pos 6. Empty seats are 2, 4, 5.
- S is opposite U: In a 6-person circle, opposite seats are 3 positions apart (e.g., 1 and 4, 2 and 5, 3 and 6).
- The empty seats are 2, 4, 5. Seat 4 is opposite Q(1). S and U must be opposite each other, so they must occupy seats 2 and 5.
- The remaining person, T, must occupy the remaining seat, which is 4.
Possible Initial Arrangements
Based on S and U occupying seats 2 and 5, two initial arrangements are possible:
- Arrangement 1: Q(1), S(2), R(3), T(4), U(5), P(6)
- Arrangement 2: Q(1), U(2), R(3), T(4), S(5), P(6)
Seating Arrangement After Seat Switch
P and U now switch seats.
- After Switch (Arrangement 1): P (at 6) and U (at 5) swap. The new clockwise order is Q(1), S(2), R(3), T(4), P(5), U(6).
- After Switch (Arrangement 2): P (at 6) and U (at 2) swap. The new clockwise order is Q(1), P(2), R(3), T(4), S(5), U(6).
Analysis of Options
We evaluate each option based on the two possible arrangements after the switch to see which statement *must* be true.
| Option | Statement | Truth Value (Arr. 1: Q,S,R,T,P,U) | Truth Value (Arr. 2: Q,P,R,T,S,U) | Must Be True? |
| 1 | P is immediately to the right of R | False (R is at 3, T is at 4) | False (R is at 3, T is at 4) | No |
| 2 | T is immediately to the left of P | True (P is at 5, T is at 4) | False (P is at 2, Q is at 1) | No |
| 3 | T is immediately to the left of P OR P is immediately to the right of Q | True (T left of P is true) OR False (P right of Q is false) = True | False (T left of P is false) OR True (P right of Q is true) = True | Yes |
| 4 | U is immediately to the right of R OR P is immediately to the left of T | False (U right of R is false) OR False (P left of T is false) = False | False (U right of R is false) OR False (P left of T is false) = False | No |
Conclusion
Option 3 is the only statement confirmed to be true in both possible seating arrangements after P and U switch places. Therefore, this statement must necessarily be true.
Important Questions from Seating Arrangement
- P and Q have been allotted a hostel room with two beds, a study table, and an almirah. P is an avid bird-watcher and wants to sit at the table and watch birds outside the window. Q does not mind that as long as his bed is close to the ceiling fan.
Which one of the following arrangements suits them the most? Six persons P, Q, R, S, T and U are sitting around a circular table facing the center not necessarily in the same order. Consider the following statements:
• P sits next to S and T.
• Q sits diametrically opposite to P.
• The shortest distance between S and R is equal to the shortest distance between T and U.
Based on the above statements, Q is a neighbor ofSeven cars P, Q, R, S, T, U and V are parked in a row not necessarily in that order. The cars T and U should be parked next to each other. The cars S and V also should be parked next to each other, whereas P and Q cannot be parked next to each other. Q and S must be parked next to each other. R is parked to the immediate right of V. T is parked to the left of U.
Based on the above statements, the only INCORRECT option given below is:- Four persons P, Q, R and S are to be seated in a row, all facing the same direction, but not necessarily in the same order. P and R cannot sit adjacent to each other. S should be seated to the right of Q. The number of distinct seating arrangements possible is:
Five 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:
