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:
Seating Arrangement Logic Puzzle
We need to find the number of distinct seating arrangements for five persons (P, Q, R, S, T) in a row based on two conditions:
- Q and R are separated by exactly one person (Q _ R or R _ Q).
- S is not seated adjacent to Q.
Understanding the Q and R Constraint
The condition 'Q and R are separated by one person' means they must be in a pattern like Q _ R or R _ Q. Let's consider the possible positions for this block of three people within the 5 available seats:
- Pattern Q _ R:
- Positions (1, 2, 3): $Q _ R _ _$
- Positions (2, 3, 4): $_ Q _ R _$
- Positions (3, 4, 5): $_ _ Q _ R$
- Pattern R _ Q:
- Positions (1, 2, 3): $R _ Q _ _$
- Positions (2, 3, 4): $_ R _ Q _$
- Positions (3, 4, 5): $_ _ R _ Q$
These are 6 possible relative placements for Q and R.
Calculating Total Arrangements (Ignoring S Constraint)
For each of the 6 placements above, the remaining 3 persons (P, S, T) need to be arranged in the 3 empty spots. The number of ways to arrange 3 distinct persons is $3! = 3 \times 2 \times 1 = 6$.
Therefore, the total number of arrangements satisfying the Q _ R / R _ Q constraint is $6 \times 6 = 36$.
Applying the 'S Not Adjacent to Q' Constraint
Now, we must subtract the arrangements where S *is* adjacent to Q from the total 36 arrangements.
Let's analyze the invalid cases for each of the 6 placements:
- $Q _ R _ _$ (Q at pos 1): S is invalid at pos 2.
- Invalid arrangements: Need S at pos 2. Remaining P, T fill pos 4, 5. $1 \times 1 \times 2! = 2$ invalid ways (e.g., QSRPT, QSRTP).
- $_ Q _ R _$ (Q at pos 2): S is invalid at pos 1 or pos 3.
- Invalid arrangements: S at pos 1 (SQ_R_). P, T fill pos 3, 5. $1 \times 2! = 2$ ways.
- Invalid arrangements: S at pos 3 (_QS R_). P, T fill pos 1, 5. $1 \times 2! = 2$ ways.
- Total invalid = $2 + 2 = 4$.
- $_ _ Q _ R$ (Q at pos 3): S is invalid at pos 2 or pos 4.
- Invalid arrangements: S at pos 2 (_SQ_R). P, T fill pos 1, 4. $1 \times 2! = 2$ ways.
- Invalid arrangements: S at pos 4 (__QS R). P, T fill pos 1, 2. $1 \times 2! = 2$ ways.
- Total invalid = $2 + 2 = 4$.
- $R _ Q _ _$ (Q at pos 3): S is invalid at pos 2 or pos 4.
- Invalid arrangements: S at pos 2 (RSQ__). P, T fill pos 4, 5. $1 \times 2! = 2$ ways.
- Invalid arrangements: S at pos 4 (R_QS_). P, T fill pos 2, 5. $1 \times 2! = 2$ ways.
- Total invalid = $2 + 2 = 4$.
- $_ R _ Q _$ (Q at pos 4): S is invalid at pos 3 or pos 5.
- Invalid arrangements: S at pos 3 (_RS Q_). P, T fill pos 1, 5. $1 \times 2! = 2$ ways.
- Invalid arrangements: S at pos 5 (_R_QS). P, T fill pos 1, 3. $1 \times 2! = 2$ ways.
- Total invalid = $2 + 2 = 4$.
- $_ _ R _ Q$ (Q at pos 5): S is invalid at pos 4.
- Invalid arrangements: Need S at pos 4. P, T fill pos 1, 2. $1 \times 2! = 2$ ways (e.g., PTRSQ, TPRSQ).
Total number of invalid arrangements (where S is adjacent to Q) = $2 + 4 + 4 + 4 + 4 + 2 = 20$.
Final Calculation
Number of distinct arrangements = (Total arrangements satisfying Q _ R/R _ Q) - (Invalid arrangements where S is adjacent to Q)
Number of distinct arrangements = $36 - 20 = 16$.
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?
