Problem Analysis
We need to find the number of ways to seat five persons (P, Q, R, S, T) in a row based on specific constraints:
- Five positions in a row: $_ _ _ _ _$
- All facing the same direction.
- Constraint 1: P and T cannot be seated at the ends (positions 1 and 5).
- Constraint 2: P should not be seated adjacent to S.
- Constraint 3: R must be in the second position from the left (position 2).
Step-by-Step Solution
Let the positions be numbered 1, 2, 3, 4, 5 from left to right.
- Place R: According to Constraint 3, R is fixed at position 2.
Arrangement: $_ R _ _ _$
- Identify possible positions for P and T: Constraint 1 states P and T cannot be at the ends (positions 1 and 5). Since R is at position 2, the available positions for P and T are {3, 4}. This means P and T must occupy positions 3 and 4.
- Identify possible positions for Q and S: The remaining positions are {1, 5} and whichever position from {3, 4} is not occupied by P or T.
- Analyze Constraint 2 (P not adjacent to S): This constraint is key to determining the valid arrangements.
Case Analysis:
We have two main possibilities for the positions of P and T in seats {3, 4}.
Case A: P is at position 3, T is at position 4.
Arrangement: $_ R P T _$
- Remaining positions for Q and S are {1, 5}.
- Check Constraint 2: P is at position 3. S cannot be adjacent to P, meaning S cannot be at position 2 (occupied by R) or position 4 (occupied by T).
- This leaves positions 1 and 5 for S. Both are valid according to this constraint relative to P.
- Subcase A1: S is at position 1. The arrangement is $S R P T _$. Q must be at position 5. Final arrangement: $S R P T Q$. This satisfies all constraints.
- Subcase A2: S is at position 5. The arrangement is $_ R P T S$. Q must be at position 1. Final arrangement: $Q R P T S$. This satisfies all constraints.
- Case A yields 2 distinct arrangements.
Case B: T is at position 3, P is at position 4.
Arrangement: $_ R T P _$
- Remaining positions for Q and S are {1, 5}.
- Check Constraint 2: P is at position 4. S cannot be adjacent to P, meaning S cannot be at position 3 (occupied by T) or position 5.
- Therefore, S must be at position 1. Position 5 is ruled out for S.
- Subcase B1: S is at position 1. The arrangement is $S R T P _$. Q must be at position 5. Final arrangement: $S R T P Q$. This satisfies all constraints.
- Subcase B2: S is at position 5. This placement is invalid because S would be adjacent to P (Constraint 2 violation).
- Case B yields 1 distinct arrangement.
Total Arrangements:
Summing the valid arrangements from both cases:
Total arrangements = (Arrangements from Case A) + (Arrangements from Case B)
Total arrangements = 2 + 1 = 3
The possible distinct seating arrangements are: $S R P T Q$, $Q R P T S$, and $S R T P Q$.
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?
