Seating Arrangement – PGECET Reasoning Ability
(Prepared for IIT‑JEE / competitive exam style questions)
1. What is a Seating Arrangement problem?
A seating‑arrangement problem asks you to determine the relative positions of a set of people (or objects) given a list of constraints.
Typical constraints involve:
| Term | Meaning | Example |
|---|---|---|
| next to / beside | Immediate neighbour (difference of 1) | A sits beside B |
| left of / right of | A is somewhere to the left / right (difference ≥ 1) | A is left of B |
| immediately left of / immediately right of | A is the direct left / right neighbour | A is immediately left of B |
| between | A is somewhere between B and C (not necessarily immediate) | A is between B and C |
| directly between | A is the immediate middle seat | A is directly between B and C |
| opposite | In a 2‑row arrangement, same column in the other row | A opposite B |
| at the end / first / last | Fixed position at the leftmost/rightmost seat | A sits at the end |
| not next to / not opposite | A cannot satisfy that relation with B | A is not next to B |
In a circular arrangement the “left”/“right” wrap around the circle.
2. Representing the problem
| Representation | Use‑case | How to write it |
|---|---|---|
| Linear row | Single row of 1…n seats | Seat 1, Seat 2, … Seat n |
| 2‑row table | Two rows of m seats (usually 5 or 6) | (Row 1, Seat 1)…(Row 1, Seat m) and (Row 2, Seat 1)…(Row 2, Seat m) |
| Circular | Circular table | Seat 1, Seat 2, … Seat n with wrap‑around |
| Grid | More than 2 rows | Use coordinates (row, column) |
Notation
pos(A) → seat number of person A
row(A) → row of person A (1 or 2)
col(A) → column of person A (1…m)
Rule of thumb: Convert every constraint into an algebraic inequality or equality involving pos, row, col.
E.g. A left of B → pos(A) < pos(B) (linear) or col(A) < col(B) (2‑row).
3. Standard Constraints & Their Algebra
| Constraint | Algebraic Form | Notes |
|---|---|---|
A is next to B |
` | pos(A) – pos(B) |
A is immediately left of B |
pos(A) + 1 = pos(B) |
For linear; in 2‑row: col(A)+1 = col(B) and row(A) = row(B) |
A is left of B |
pos(A) < pos(B) |
No adjacency requirement |
A is between B and C |
pos(B) < pos(A) < pos(C) or vice‑versa |
Order of B & C not specified |
A is directly between B and C |
pos(A) = (pos(B)+pos(C))/2 (must be integer) |
In 2‑row: same column, opposite row |
A is opposite B |
col(A) = col(B) and row(A) ≠ row(B) |
In 2‑row only |
A at end |
pos(A)=1 or pos(A)=n |
For linear; in 2‑row: col(A)=1 or col(A)=m |
A **not next to |