Total Possible Arrangements
The total number of distinct ways to arrange four persons (P, Q, R, S) in a row is given by the permutation formula $P(n, n) = n!$.
For 4 persons, the total arrangements are $4! = 4 \times 3 \times 2 \times 1 = 24$. Let this be $N$.
Applying Constraints with Inclusion-Exclusion Principle
We need to find arrangements where two conditions are met simultaneously: 1. P and R are NOT adjacent. 2. S is seated to the RIGHT of Q. We use the inclusion-exclusion principle. Let $A$ be the set of arrangements where P and R ARE adjacent. Let $B$ be the set of arrangements where S is to the LEFT of Q. We want to find the number of arrangements that are neither in $A$ nor in $B$. This is given by $N - |A \cup B| = N - (|A| + |B| - |A \cap B|)$.
Size of Set A: P and R Adjacent ($|A|$)
- Treat 'PR' as a single block. The entities to arrange are (PR), Q, S. This gives $3! = 6$ arrangements.
- Treat 'RP' as a single block. The entities to arrange are (RP), Q, S. This gives $3! = 6$ arrangements.
- The total number of arrangements where P and R are adjacent is $|A| = 6 + 6 = 12$.
Size of Set B: S to the Left of Q ($|B|$)
- In any arrangement of 4 distinct items, exactly half will have S to the left of Q, and the other half will have S to the right of Q.
- The number of arrangements where S is to the left of Q is $|B| = N / 2 = 24 / 2 = 12$.
Size of Intersection $A \cap B$: P, R Adjacent AND S Left of Q ($|A \cap B|$)
- Case 1: Block 'PR'. Permutations of (PR), Q, S are 6. Those where S is left of Q are: (PR)SQ, S(PR)Q, SQ(PR). (3 arrangements).
- Case 2: Block 'RP'. Permutations of (RP), Q, S are 6. Those where S is left of Q are: (RP)SQ, S(RP)Q, SQ(RP). (3 arrangements).
- Total arrangements in the intersection is $|A \cap B| = 3 + 3 = 6$.
Calculating $|A \cup B|$
- Using the inclusion-exclusion formula:
- $|A \cup B| = |A| + |B| - |A \cap B|$
- $|A \cup B| = 12 + 12 - 6 = 18$.
- This is the number of arrangements where P and R are adjacent OR S is to the left of Q (or both).
Final Calculation: Arrangements Satisfying Both Conditions
- We need arrangements where P and R are NOT adjacent AND S is to the RIGHT of Q. This is the complement of $A \cup B$.
- Number of valid arrangements = $N - |A \cup B|$
- Number of valid arrangements = $24 - 18 = 6$.
Important Questions from Seating Arrangement
Eight students (P, Q, R, S, T, U, V, and W) are playing musical chairs. The figure indicates their order of position at the start of the game. They play the game by moving forward in a circle in the clockwise direction. After the 1st round, 4th student behind P leaves the game. After 2nd round, 5th student behind Q leaves the game. After 3rd round, 3rd student behind V leaves the game. After 4th round, 4th student behind U leaves the game. Who all are left in the game after the 4th round?
Note: The figure shown is representative.
- 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:- P, Q, R, S, T, U, V, and W are seated around a circular table.
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?
