I. S is seated opposite to W.
II. U is seated at the second place to the right of R.
III. T is seated at the third place to the left of R.
IV. V is a neighbour of S.
Which of the following must be true?
1. Setting up the Arrangement
We are arranging 8 people (P, Q, R, S, T, U, V, W) around a circular table. Let's number the seats 1 to 8 clockwise. Opposite pairs of seats are (1,5), (2,6), (3,7), and (4,8).
2. Applying the Conditions
- Condition I: S is opposite W. We can place S at seat 1 and W at seat 5.
- Conditions II & III: U is 2nd to the right of R (position $R+2$). T is 3rd to the left of R (position $R-3$ or $R+5$). This implies a relative block: T _ _ R _ U _ _ in clockwise order.
- Condition IV: V is a neighbour of S. Since S is at seat 1, V must be at seat 2 or seat 8.
3. Analyzing Possible Arrangements
We test placements for R based on the T _ _ R _ U _ _ block, considering S(1) and W(5):
- Scenario 1: R is at seat 2.
- U is 2nd right of R(2) $\rightarrow$ U is at seat 4.
- T is 3rd left of R(2) $\rightarrow$ T is at seat 7 (Clockwise sequence: 2 $\rightarrow$ 1 $\rightarrow$ 8 $\rightarrow$ 7).
- Current placement: S(1), R(2), _(3), U(4), W(5), _(6), T(7), _(8).
- V must neighbour S(1). Neighbours are 8 and 2(R). V must be at seat 8.
- Final arrangement for this scenario: S(1), R(2), _(3), U(4), W(5), _(6), T(7), V(8).
- The remaining seats are 3 and 6. P and Q must occupy these seats.
- Are P and Q opposite? P/Q are at seats 3 and 6. Seat 3 is opposite seat 7. Seat 6 is opposite seat 2. Thus, P and Q are not opposite each other.
- Scenario 2: R is at seat 6.
- U is 2nd right of R(6) $\rightarrow$ U is at seat 8.
- T is 3rd left of R(6) $\rightarrow$ T is at seat 3 (Clockwise sequence: 6 $\rightarrow$ 5 $\rightarrow$ 4 $\rightarrow$ 3).
- Current placement: S(1), _(2), T(3), _(4), W(5), R(6), _(7), U(8).
- V must neighbour S(1). Neighbours are 8(U) and 2(?). V must be at seat 2.
- Final arrangement for this scenario: S(1), V(2), T(3), _(4), W(5), R(6), _(7), U(8).
- The remaining seats are 4 and 7. P and Q must occupy these seats.
- Are P and Q opposite? P/Q are at seats 4 and 7. Seat 4 is opposite seat 8. Seat 7 is opposite seat 3. Thus, P and Q are not opposite each other.
- Other placements for R (e.g., R=3, R=4, R=7, R=8) lead to contradictions with the given conditions (like U being placed opposite S or W, or T's position being occupied). Both valid scenarios derived above are consistent with all conditions.
4. Conclusion: Statement That Must Be True
In both consistent seating arrangements derived from the conditions:
- P and Q occupy seats that are not opposite each other.
- Therefore, the statement "P is not seated opposite to Q" must 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:
- P, Q, R, S, T and U are seated around a circular table. R is seated two places to the right of Q. P is seated three places to the left of R. S is seated opposite U. If P and U now switch seats, which of the following must necessarily be true?
