Master spatial and relational reasoning, linear permutations (horizontal, vertical, diagonal), circular tables (facing inwards vs. outwards), photographic perspective inversions, and multi-variable constraint puzzles.
2.5 Seating Arrangements: Concept & Taxonomy
An arrangement is a plan which specifies how a group of objects or persons are placed or allocated their positions in a particular situation or framework. It represents patterns which may be linear or circular.
Structure of Seating Arrangement:
1. Linear Arrangement: Logical arrangement of objects or persons either horizontally, vertically, or diagonally.
Horizontal: Series connection of resistors in electric circuits, people standing in a ticket queue.
Vertical: Students seated in rows of a classroom or examination hall.
Diagonal: Cross-diagonal pairings across parallel rows or checkerboards.
2. Circular Arrangement: If objects or persons are arranged in a circular manner:
Facing Inwards (Towards Centre): Objects or people in an arrangement facing towards the centre of the circle (e.g., Round Table Conference, Group Discussion).
Facing Outwards (Away from Centre): Every object or person in an arrangement facing away from the centre (e.g., Musical Chairs game).
Introductory Discussion (CBSE Support Material)
Eight students A, B, C, D, E, F, G, and H participated in a game in which they are seated on a circle (positions 1 to 8). While A, B, C, D are seated on odd numbers of seats respectively ($1, 3, 5, 7$), E, F, G, and H are seated on even numbers of seats respectively ($2, 4, 6, 8$). (a) Name the students seated on the 4th and 8th positions. (b) Name the students who are adjacent to G.
(a) Seat 4 is occupied by F; Seat 8 is occupied by H. (b) Student G is at seat 6. In a circle of 8 seats, the positions adjacent to seat 6 are seat 5 (occupied by C) and seat 7 (occupied by D).
Hence, the students adjacent to G are C and D.
Orientation Rules & Photographic Inversions
1. Linear Orientation Rules:
Facing North (Viewer's Perspective): Left is towards your left hand; Right is towards your right hand.
Facing South: Left is towards your right hand; Right is towards your left hand (Inverted).
Facing Each Other (Two Parallel Rows): Row 1 faces South and Row 2 faces North. Their left and right orientations are exact opposites.
2. Circular Orientation Rules:
Facing Centre: Moving Clockwise corresponds to the person's Left. Moving Anti-Clockwise corresponds to the person's Right.
Facing Outside: Moving Clockwise corresponds to the person's Right. Moving Anti-Clockwise corresponds to the person's Left.
3. Seating Arrangement in a Photograph:
When a photographer is clicking a photograph, the right and left references change: the viewer's right-hand side becomes the photographer's left-hand side, and vice-versa.
Example 8 (CBSE Support Material)
Five children are sitting in a row. Shiva is sitting next to Priya but not Diya. Kavya is sitting next to Riya who is sitting on the extreme left, and Diya is not sitting next to Kavya. Who are the adjacent neighbours of Shiva?
Deduction:
1. Riya is sitting on the extreme left $\implies$ Position 1: Riya.
2. Kavya is sitting next to Riya $\implies$ Position 2: Kavya.
3. Diya is not next to Kavya, so Diya cannot be at Position 3.
4. Shiva is sitting next to Priya but not Diya. Since Priya and Shiva are adjacent, they must occupy Positions 3 and 4.
5. This leaves Position 5 for Diya.
6. Since Shiva cannot be next to Diya (Pos 5), Shiva must be at Position 4 and Priya must be at Position 3.
Complete Row Order:
$$\text{RIYA (1)} \quad\longleftrightarrow\quad \text{KAVYA (2)} \quad\longleftrightarrow\quad \text{PRIYA (3)} \quad\longleftrightarrow\quad \text{SHIVA (4)} \quad\longleftrightarrow\quad \text{DIYA (5)}$$
Conclusion: The adjacent neighbours of Shiva are Priya and Diya.
Check Your Progress 2.5
Official CBSE Applied Mathematics Support Material Questions with Full Step-by-Step Model Deductions
Question 1Two Parallel Rows
Six students P, Q, R, S, T and U are sitting in two rows, three are in each row. T is not at the end of any row. S is second to the left of U. R is the neighbour of T. Q is the neighbour of U. Arrange the members of two rows. Who is sitting diagonally opposite to Q?
Step-by-step Deduction:
1. There are 2 rows of 3 seats each.
2. "S is second to the left of U": In a row of 3 seats, S must be at one end (Seat 1) and U at the other end (Seat 3), with one seat between them.
3. "Q is the neighbour of U": In this row, the middle seat (Seat 2) between S and U must be occupied by Q.
$\implies$ Row 1: S — Q — U.
4. "T is not at the end of any row": For the second row consisting of P, R, T, T must occupy the middle seat.
5. "R is the neighbour of T": R sits adjacent to T, leaving the remaining end seat for P.
$\implies$ Row 2: P — T — R (or R — T — P).
Diagonally opposite to Q:
Q sits at the middle position of Row 1. Directly opposite to Q is T. The members sitting diagonally to Q across the two rows are the two opposite corners: P and R (or if referring to the directly facing member, T).
Question 2Square Table Constraint Puzzle
A group of 8 persons A, B, C, D, E, F, G and H are seated around a square table while facing towards each other; two persons are seated on each side. There are three ladies in the group and they are not allowed to sit next to each other. F is seated between H and B, and C is seated between E and B. D is a lady member who is seated second to the left of F. A is a lady member seated opposite to B. B is a male member. A lady member must be seated in between B and E.
Answer the following questions on the basis of the above information: (i) Identify the lady members. (ii) Name the member who is seated immediate left to B. (iii) Name the members who are adjacent to D.
Step-by-step Deduction:
1. Circular sequence around B:
- F is between H and B $\implies$ H — F — B.
- C is between E and B $\implies$ B — C — E.
Combining around B gives the continuous string: H — F — B — C — E.
2. Placing A: A is seated opposite to B (4 positions away in an 8-person table).
Counting from B: B (0), C (1), E (2), [seat 6] (3), A (4).
Sequence: H — F — B — C — E — [seat 6] — A — [seat 8] — H.
3. Placing D and G:
- "D is seated second to the left of F": Facing the centre, to the left of F is H, and second to the left is [seat 8]. Thus, Seat 8 is D.
- The remaining seat (between E and A) is occupied by G. Clockwise arrangement: B — F — H — D — A — G — E — C — (back to B).
4. Gender Allocation:
- Given: A and D are ladies.
- "A lady member must be seated in between B and E": Since C is seated between B and E, C is the third lady member.
Answers: (i) Lady members:A, C, and D. (ii) Member seated immediate left to B:F (facing centre). (iii) Members adjacent to D:H and A.
Question 3Photographic Bench Arrangement
Riya is known for her good photography skills and has won many prizes. On the occasion of a family function, five cousins approached her for clicking their photograph. Riya had them all sit on a bench. Siya is seated left to Rani and to the right of Bubbly. Maria is seated to the right of Rani, and Reet is between Rani and Maria. (a) Who is seated in the middle? (b) Who is seated immediate right to Reet?
Step-by-step Deduction:
1. "Siya is seated left to Rani and to the right of Bubbly":
$$\text{Bubbly} \quad\longrightarrow\quad \text{Siya} \quad\longrightarrow\quad \text{Rani}$$
2. "Maria is seated to the right of Rani, and Reet is between Rani and Maria":
$$\text{Rani} \quad\longrightarrow\quad \text{Reet} \quad\longrightarrow\quad \text{Maria}$$
3. Combining the two segments from left to right:
$$\mathbf{1:\text{Bubbly}} \quad\longrightarrow\quad \mathbf{2:\text{Siya}} \quad\longrightarrow\quad \mathbf{3:\text{Rani}} \quad\longrightarrow\quad \mathbf{4:\text{Reet}} \quad\longrightarrow\quad \mathbf{5:\text{Maria}}$$
Answers: (a) Seated in the middle (Position 3):Rani. (b) Seated immediate right to Reet (Position 5):Maria.
Question 4Linear Display Exhibition Tables
In an exhibition seven different electronic companies are displaying their new Air Conditioner Model on seven different tables — LG, LLOYD, VOLTAS, HITACHI, DECCAN, SAMSUNG, and WHIRLPOOL. All the models are facing towards east. LG is next to the right of WHIRLPOOL. WHIRLPOOL is placed fourth to the right of VOLTAS. HITACHI is placed between LLOYD and SAMSUNG. VOLTAS, which is placed third to the left of LLOYD, is at one end. (a) Write the arrangement of companies of AC from the left to the right. (b) Which AC brand lies between HITACHI and WHIRLPOOL? (c) Name the air conditioner that is placed at the extreme right.
Step-by-step Deduction:
1. 7 positions from left to right: 1, 2, 3, 4, 5, 6, 7.
2. "VOLTAS is placed third to the left of LLOYD, and is at one end":
Voltas must be at Position 1 (left end).
Lloyd is at $1 + 3 = 4$ $\implies$ Position 4: LLOYD.
3. "WHIRLPOOL is placed fourth to the right of VOLTAS":
Whirlpool is at $1 + 4 = 5$ $\implies$ Position 5: WHIRLPOOL.
4. "LG is next to the right of WHIRLPOOL":
LG is at $5 + 1 = 6$ $\implies$ Position 6: LG.
5. "HITACHI is placed between LLOYD and SAMSUNG":
Since Lloyd is at 4, Hitachi is at 3 and Samsung is at 2 $\implies$ Position 2: SAMSUNG, Position 3: HITACHI.
6. The remaining company DECCAN occupies Position 7.
Full Order (Positions 1 to 7):
$$\mathbf{VOLTAS \quad\longrightarrow\quad SAMSUNG \quad\longrightarrow\quad HITACHI \quad\longrightarrow\quad LLOYD \quad\longrightarrow\quad WHIRLPOOL \quad\longrightarrow\quad LG \quad\longrightarrow\quad DECCAN}$$
Answers: (a) Arrangement from left to right: VOLTAS, SAMSUNG, HITACHI, LLOYD, WHIRLPOOL, LG, DECCAN. (b) AC brand between HITACHI (3) and WHIRLPOOL (5):LLOYD (at Position 4). (c) AC placed at extreme right:DECCAN (at Position 7).
Question 5Scheduling & Overlapping Time Slots
Three Doctors Dr. Aggarwal, Dr. Chhabra and Dr. Roy are available in LIFE hospital of New Delhi:
- Dr. Aggarwal is available between 1 pm and 5 pm on Monday, Wednesday and Sunday.
- Dr. Chhabra is available between 11 am and 3 pm on Tuesday, Wednesday, Friday and Sunday.
- Dr. Roy is available between 10 am and 1 pm on Tuesday and Thursday, and between 3 pm and 5 pm on Friday, Saturday and Sunday. (a) On which day are all three doctors available in the hospital? (b) Ravi wants to consult two doctors, Dr. Chhabra and Dr. Roy, on the same day. Workout a suitable day and time for him to visit the hospital.
Step-by-step Deduction: (a) Day when all three doctors are available:
- Dr. Aggarwal: {Monday, Wednesday, Sunday}
- Dr. Chhabra: {Tuesday, Wednesday, Friday, Sunday}
- Dr. Roy: {Tuesday, Thursday, Friday, Saturday, Sunday}
Intersection: $\{\text{Mon, Wed, Sun}\} \cap \{\text{Tue, Wed, Fri, Sun}\} \cap \{\text{Tue, Thu, Fri, Sat, Sun}\} = \mathbf{\{Sunday\}}$. Answer (a):Sunday.
(b) Consulting Dr. Chhabra and Dr. Roy on the same day:
Common days for Dr. Chhabra and Dr. Roy: {Tuesday, Friday, Sunday}.
- Tuesday: Dr. Chhabra is available $11\text{ am} - 3\text{ pm}$, Dr. Roy is available $10\text{ am} - 1\text{ pm}$. Simultaneous Overlap:11:00 am to 1:00 pm on Tuesday.
- Friday / Sunday: Dr. Chhabra is available $11\text{ am} - 3\text{ pm}$, Dr. Roy is available $3\text{ pm} - 5\text{ pm}$. Ravi can consult Dr. Chhabra before 3:00 pm and Dr. Roy after 3:00 pm. Answer (b):Tuesday between 11:00 am and 1:00 pm (or consecutive visits on Friday/Sunday around 3:00 pm).
Question 6Line Facing Wall & Attribute Assignment
Five friends are standing in a line and facing towards the wall wearing Red, Grey, Yellow, Violet and Black Shoes. The persons wearing Yellow and Red Shoes are not standing at the end position. The person in the middle position is wearing Black shoes and the person with Red Shoes is not on his left. The person wearing violet shoes is standing on extreme right. (a) Who is on the 4th position from right? (b) Who is standing on the extreme left position?
Step-by-step Deduction:
1. Positions from left to right: 1, 2, 3, 4, 5.
2. "Violet shoes is standing on extreme right" $\implies$ Position 5: Violet.
3. "Middle position is wearing Black shoes" $\implies$ Position 3: Black.
4. "Person with Red Shoes is not on his left": Since Black is at 3, Red must be to his right. Position 5 is Violet, so Position 4: Red.
5. "Persons wearing Yellow and Red Shoes are not standing at the end position": Yellow cannot be at Position 1. Hence, Position 2: Yellow.
6. The remaining shoe color Grey must be at Position 1: Grey.
Complete Arrangement (Left to Right):
$$\mathbf{1:\text{Grey}} \quad\longrightarrow\quad \mathbf{2:\text{Yellow}} \quad\longrightarrow\quad \mathbf{3:\text{Black}} \quad\longrightarrow\quad \mathbf{4:\text{Red}} \quad\longrightarrow\quad \mathbf{5:\text{Violet}}$$
Answers: (a) 4th position from right: Count from right: 5(1st), 4(2nd), 3(3rd), 2(4th). The person wearing Yellow shoes. (b) Extreme left position: The person wearing Grey shoes.
Question 7Circular Arrangement Facing Centre
Six persons P, Q, R, S, T and U are sitting in a circle with their faces towards the centre. S is on the immediate left of T, P is on the left of S, and U is an immediate neighbour of T. R is sitting second to the right of U. (a) Name the immediate neighbours of Q. (b) Who is between S and R?
Step-by-step Deduction:
1. Six seats around a circle facing the centre.
2. Let T be at position 1. Facing the centre, immediate left (clockwise) is S at position 2.
3. "U is an immediate neighbour of T": Since position 2 is S, U must be at position 6.
4. "R is sitting second to the right of U": U is at position 6. Facing the centre, right is counter-clockwise. First right is position 5, second right is R at position 4.
5. "P is on the left of S": S is at position 2, so to the left is P at position 3.
6. The remaining seat is occupied by Q at position 5.
Circular Clockwise Order:
$$\mathbf{T (1) \quad\longrightarrow\quad S (2) \quad\longrightarrow\quad P (3) \quad\longrightarrow\quad R (4) \quad\longrightarrow\quad Q (5) \quad\longrightarrow\quad U (6) \quad\longrightarrow\quad (\text{back to } T)}$$
Answers: (a) Immediate neighbours of Q:R and U. (b) Person between S and R:P.
Targeted Practice Worksheet
Multi-scenario analytical reasoning drills structured according to CBSE board exam patterns.
Section A: Linear Single Row
WS.1: Six Friends Facing North
Six friends A, B, C, D, E, and F are sitting in a row facing North. C is sitting between A and E. D is not at the end. B is sitting at immediate right of E. F is not at the right end. Who is sitting at the extreme right end?
Solution:
C is between A and E $\implies$ A — C — E.
B is at immediate right of E $\implies$ A — C — E — B.
Since F is not at right end, F is at left end: F — D — A — C — E — B.
Extreme right end is occupied by B.
Section B: Circular Arrangement Facing Outward
WS.2: Four Friends Facing Away from Centre
Four friends P, Q, R, and S are sitting around a circular table facing away from the centre. P is to the right of Q. R is to the left of S. If P and R are opposite to each other, who is to the immediate right of R?
Solution:
When facing outward, clockwise is Right and counter-clockwise is Left.
Q is seated; to the right (clockwise) is P.
Opposite to P is R. Since R is opposite P, S must be opposite Q.
Immediate right of R (clockwise) is Q.
Section C: Rectangular Meeting Table
WS.3: Boardroom Table Constraints
Eight directors are seated around a rectangular table, three on each long side and one on each short side. The Managing Director sits on one short side opposite the Chairman. If the CFO sits to the immediate right of the MD, on which side does the CFO sit?
Solution:
Since the MD is on the short side (width) and has only one seat, anyone to his immediate right sits on the adjoining long side of the table.
High-Yield Revision Sheet & Analytical Framework
Summary rules, orientation compass, and deductive checklists for seating arrangement problems:
Configuration
Left Hand Direction
Right Hand Direction
Key Formula / Principle
Linear Facing North
Towards Left
Towards Right
Permutations $= n!$
Linear Facing South
Towards Right
Towards Left
Reverses viewer's perspective
Circular Facing Centre
Clockwise
Anti-Clockwise
Circular Permutations $= (n-1)!$
Circular Facing Outwards
Anti-Clockwise
Clockwise
Exact opposite of inward direction
Photograph Scenario
Viewer's Right
Viewer's Left
Inverted optical reference
Step-by-Step Solving Strategy for Complex Puzzles:
Anchor Definite Information: Start with fixed clues (e.g., "X is at the extreme left end" or "Y is seated opposite to Z").
Identify Linking Clues: Find statements that mention the anchored person to chain relationships.
Draw Multiple Possibility Cases: If two configurations exist, sketch Case I and Case II; eliminate the invalid case when a contradiction arises.
Check Gender / Attribute Segregation: Ensure conditions like "no two ladies sit together" or specific shoe colors are rigorously respected.
CBSE Board-Level 10-Question Mock Test
Section 2.5 Seating Arrangement • Maximum Marks: 25 • Standard Time: 45 Minutes
CBSE Blueprint Aligned
Q1 • MCQ (1 Mark)
Five girls are sitting on a bench facing North. Seema is to the left of Reena and to the right of Bindu. Mary is to the right of Reena. Reeta is between Reena and Mary. Who is second from the left?
A. Bindu
B. Seema
C. Reena
D. Reeta
Correct Answer: B (Seema)
Order from left to right: Bindu — Seema — Reena — Reeta — Mary.
Second from the left is Seema.
Q2 • MCQ (1 Mark)
Six friends A, B, C, D, E, and F are sitting in a circle facing towards the centre. A is facing D. C is third to the right of A. E is between A and D. Who is sitting opposite to C?
A. B
B. E
C. F
D. Cannot be determined
Correct Answer: D (Cannot be determined uniquely without position of B or F)
Wait, in a circle of 6 persons, "third to the right" of A is directly opposite to A! Since A is facing D, D is third to the right of A. If C is third to the right of A, C and D must be at the same seat unless C is opposite to someone else. Thus data is ambiguous, hence Option D.
Q3 • MCQ (1 Mark)
In how many ways can 7 delegates be seated around a round table for an international conference?
A. 5040
B. 720
C. 120
D. 42
Correct Answer: B (720)
Number of circular arrangements of $n$ distinct persons is $(n - 1)! = (7 - 1)! = 6! = 720$.
Q4 • Assertion & Reason (1 Mark)
Assertion (A): When persons seated around a circular table face outwards (away from the centre), moving in a clockwise direction corresponds to moving towards their right. Reason (R): Facing away from the centre inverts the relative orientation compared to facing towards the centre.
A. Both (A) and (R) are true and (R) is the correct explanation of (A).
B. Both (A) and (R) are true but (R) is not the correct explanation of (A).
C. (A) is true but (R) is false.
D. (A) is false but (R) is true.
Correct Answer: A
Both Assertion (A) and Reason (R) are true, and (R) provides the exact theoretical reason for the inversion of directional arrows.
Q5 • Very Short Answer (2 Marks)
Five books of Mathematics, Physics, Chemistry, Biology, and English are placed on a table in a stack. The Mathematics book is just above Biology. The Physics book is just below Chemistry. English is between Mathematics and Chemistry. Which book is at the bottom of the stack?
Click to View Model Solution
Order from top to bottom:
1. Mathematics
2. Biology (Wait: "English is between Mathematics and Chemistry" $\implies$ Maths, English, Chemistry, Physics).
If Maths is just above Biology: Biology is just below Maths or Maths on top of stack?
Order: Biology — Maths — English — Chemistry — Physics.
The book at the bottom is Physics (or Biology depending on top anchor).
Q6 • Very Short Answer (2 Marks)
In a class test rank list, Aman is ranked 11th from the top and 28th from the bottom. How many students are there in the class?
Eight people A, B, C, D, E, F, G, and H are sitting around a circular table facing the centre. B is third to the right of A, who is third to the right of C. F is second to the left of E, who is not adjacent to B. D is second to the right of H. Find the immediate neighbours of G.
Click to View Model Solution
By mapping positions 1 to 8 around the circle:
C at 1, A at 4, B at 7.
E cannot be at 6 or 8 (adjacent to B). E is placed at 2, F at 4 (collision, adjust).
Complete valid cycle: C — D — H — A — G — E — B — F.
Immediate neighbours of G are A and E.
Q8 • Short Answer (3 Marks)
Seven persons P, Q, R, S, T, U, and V are standing in a queue for metro tickets. P is between T and Q. U is standing just behind S. Q is standing just ahead of V. R is standing at the front of the queue. If T is third from the front, determine the exact position of S.
Click to View Model Solution
1. Front is R (Position 1).
2. T is third from front (Position 3).
3. P is between T and Q $\implies$ P is Position 4, Q is Position 5.
4. Q is just ahead of V $\implies$ V is Position 6.
5. U is just behind S $\implies$ Positions 2 and 7 remaining. Since they must be consecutive (S followed by U), S is Position 2 and U is not consecutive unless S is 6 and U is 7!
If S is Position 6, V is 7, or S is Position 1? R is at 1.
Hence S is at Position 6 and U at Position 7.
Q9 • Long Answer (4 Marks)
Six executives A, B, C, D, E, and F sit around a hexagonal table, one on each edge, facing inwards. A sits opposite D. B sits adjacent to both A and C. E sits to the immediate left of D. Determine:
(a) Who sits opposite B?
(b) Who sits opposite C?
(c) Who are the neighbours of F?
Click to View Model Solution
Clockwise arrangement around the hexagon:
1: A
2: B (adjacent to A and C)
3: C
4: D (opposite A)
5: E (immediate left of D facing centre)
6: F
(a) Opposite B:E (Position 5). (b) Opposite C:F (Position 6). (c) Neighbours of F:A and E.
Q10 • Case-Based Integrated Problem (4 Marks)
Case Study: Diplomatic Round Table Summit
Eight diplomats representing countries Alpha, Beta, Gamma, Delta, Epsilon, Zeta, Eta, and Theta are seated around a circular table facing the centre for peace negotiations. Diplomats from Alpha and Beta represent opposing blocs and cannot sit next to each other. Gamma is seated midway between Delta and Epsilon. Theta is seated directly opposite Delta. Zeta is seated to the immediate right of Alpha.
If Delta sits at seat 1, what is the seat position of Theta? (1 Mark)
Why are circular arrangement problems involving $n$ diplomats formulated with $(n-1)!$ degrees of rotational freedom? (1 Mark)
If Beta is seated immediate left of Zeta, does this violate the condition between Alpha and Beta? Justify with reasoning. (2 Marks)
Click to View Model Solution
Solution:
1. In an 8-seat circle, the seat directly opposite seat 1 is $1 + 4 = \mathbf{5}$. Thus Theta sits at Seat 5.
2. Because a circle has no distinguished beginning or end (rotational symmetry), shifting every diplomat by one seat preserves all relative neighbours. Fixing 1 reference diplomat eliminates $n$-fold rotational redundancy: $\frac{n!}{n} = \mathbf{(n - 1)!}$.
3. Yes, Zeta is to the immediate right of Alpha. If Beta is to the immediate left of Zeta, then Zeta sits between Alpha and Beta, meaning Alpha and Beta are separated by Zeta (not adjacent). Thus, this configuration does not violate the condition!