Seating Arrangements notes

MDCAT Analytical and Logical Reasoning

Seating arrangement problems test the ability to place people or objects in rows, circles, facing directions, opposite positions, blocks and vertical stacks. The main skills are reading positional words correctly, changing left and right according to direction, fixing definite positions, and counting valid arrangements without double counting.

Basic Language of Position

A seating arrangement describes the position of people or objects relative to one another. Read every statement carefully and convert it into a fixed order or position. Words such as immediate, extreme, opposite, between, ahead and behind give exact information.

In a row, the reader normally views the arrangement from left to right. If people face north, their own left and right agree with the reader's left and right. If they face south, their own left and right are reversed when compared with the reader's view.

Immediate left or right means directly adjacent, with no person between.
Extreme left and extreme right mean the first and last positions in a row as seen by the reader.
Between two people means a person or persons lie in the positions separating them.
Ahead means in front of a person in the direction of movement. Behind means following that person.
In a vertical stack, above means higher and below means lower.
The order C, B, A, D, E has A exactly in the middle because it is the third of five positions.

Linear Arrangements and Fixed Positions

For a row, draw numbered places from left to right. Then place every person whose position is fixed. Use the remaining clues to fill the empty places. Numbering prevents confusion between a person's position from the left and from the right.

If a person is rth from the left and sth from the right, the total number of people is r + s - 1. The position is counted twice, so one is subtracted.

Position from the left plus position from the right equals total people plus one.
If Bilal is 5th from the left and Sara is 5th from the right in a row of 12, Sara is 8th from the left.
Bilal is at position 5 and Sara is at position 8, so two people sit between them, at positions 6 and 7.
If Ahmed is 4th from the left and 3rd from the right, total people are 4 + 3 - 1 = 6.
A statement may contain an inconsistent total. The position clues determine the total when they clearly conflict with a given number.
If R is in seat 2 and S is three seats to the right, S is in seat 5.

Direction and Point of View

Direction must be identified before interpreting left and right. A person facing north sees the same left and right as the reader. A person facing south sees the opposite left and right. The phrase from the reader's point of view refers to the diagram, while from the person's own point of view refers to the direction in which that person faces.

For example, if W is at the extreme left as seen by a reader and W faces south, W's own right side points toward the reader's left. Therefore, W is at W's own extreme right.

Facing north: reader's left is the person's left, and reader's right is the person's right.
Facing south: reader's left is the person's right, and reader's right is the person's left.
If X is immediately left of Y from the reader's view in a north-facing row, X is also on Y's left.
If people face south, reverse left and right before answering from their own point of view.
In a circular arrangement facing the centre, immediate right and immediate left are determined from each person's own viewpoint.
Do not change the actual diagram when changing viewpoint. Change only the labels left and right.

Rows with Blocks and Restrictions

Some clues join people into a fixed block. If M is immediately left of N and O is immediately right of N, the three-person block is MNO. The block can be placed at either end of a row of four if P is the remaining person.

Restrictions can be handled by counting all arrangements first and then removing the arrangements that break the restriction. This is called the complement method.

For six distinct students in a row, the total arrangements are 6! = 720.
If A and B must not sit next to each other, count total arrangements and subtract arrangements in which A and B are together.
For the together case, treat A and B as one block. The block can be AB or BA, so its two internal orders must be included.
A block of three fixed people occupies consecutive positions and moves as one unit when its order is fixed.
In M, N, O with M immediately left of N and O immediately right of N, P can be placed at either end of the block.
If Q is at the extreme left, T at the extreme right, and R immediately right of Q, the order begins Q, R and ends with T. P or S can occupy the middle position.

Adjacency, Ends and Middle Positions

Words such as immediately, next to, adjacent and beside describe neighbouring positions. A person's possible locations can often be found by first removing forbidden positions, such as the two ends.

In a row of an odd number of places, there is one middle position. In a row of an even number of places, there are two central positions, unless the question defines a different meaning.

In a row of five, a person who cannot sit at either end can occupy positions 2, 3 or 4, so there are three possible seats.
The first and last positions are the two end positions of a row.
For five positions, position 3 is the exact middle.
For the order C, B, A, D, E, the third position is occupied by A.
If a person is immediately to the right of another, no empty seat or person may occur between them.
Always check whether an answer asks for a definite position or merely a possible position.

Facing Rows and Opposite Seats

In two rows facing each other, a person in one row is opposite the person directly across from them in the other row. Draw the rows parallel to each other and align the seats in columns.

If A sits in the first row and faces B, B is in the second row directly opposite A. The direction faced may affect left and right, but it does not change which seat is opposite.

Two rows of three facing each other contain three opposite pairs.
The person directly across is called the facing or opposite person.
If the rows face each other, the left side of one row may correspond to the right side of the other row from the reader's view.
Use vertical lines to match opposite seats before applying left or right clues.
A person facing north and a person facing south can still be opposite if their seats are aligned.
Do not treat a diagonal seat as directly opposite.

Circular Arrangements

In a circular arrangement, there is no fixed extreme left or extreme right. Rotating the entire arrangement does not create a new arrangement when rotations are counted as the same. To solve a circular problem, fix one person at a reference position and arrange the others relative to that person.

For people facing the centre, clockwise and anticlockwise directions are read according to the arrangement. Immediate right and immediate left are defined from each person's own viewpoint, so the reader's visual direction must be checked carefully.

The number of circular arrangements of n distinct people, when rotations are the same, is (n - 1)!
Four people around a circular table have (4 - 1)! = 3! = 6 distinct arrangements.
Eight people around a circle have seven other relative positions after one person is fixed.
With eight people, a person and the person opposite them have three people between them on each side.
With six people, moving three places clockwise reaches the opposite person.
If P is opposite S and Q is immediately right of P, the person immediately left of S cannot be determined unless the remaining positions are specified.

Square Tables and Position Types

A square table has four sides and four corners. If two people sit on each side, the stated positions are side positions, not corner positions. The seats may be located along the sides without occupying the corners.

Always distinguish between a round table and a square table. A round table has no corners, while a square table has four corner positions.

Eight people with two on each side of a square table occupy side positions.
If no person is assigned to a corner, the number sitting at corner positions is zero.
A person at a corner touches two sides of a square table.
A person sitting at the middle of a side is not at a corner.
For a circular table, opposite positions exist when the number of seats is even.
For an odd number of seats around a circle, there is no seat exactly opposite a given seat.

Vertical, Movement and Counting Problems

Not every arrangement is horizontal. Books may be stacked vertically, and people or animals may move along a line. The same method applies: translate the statements into an order and follow the direction of movement.

For a pursuit problem, first write the order of the dog and the children. If Max is ahead of the dog and moving in the opposite direction, he passes the group, turns around and joins the pursuit. After turning, he runs behind the dog, so he is the last in the pursuit line.

If the physics book is above the chemistry book and biology is below the chemistry book, biology is lowest among the three.
Above and below refer to vertical order, not left and right.
In a pursuit line, the front position is ahead of the dog and the last position is behind all following children.
When a person changes direction, reassess ahead and behind after the turn.
For three distinct people in a row, the number of orders is 3! = 6.
Factorial notation means n! = n × (n - 1) × ... × 2 × 1.

Key terms

Linear arrangement: An arrangement in a straight row from left to right.
Circular arrangement: An arrangement around a circle in which there are no extreme end positions.
Immediate neighbour: A person sitting directly next to another person with no one between them.
Extreme position: The first or last place in a linear row.
Opposite position: The seat directly across from a person in a facing row or suitable circular arrangement.
Block: Two or more people treated as one unit because they must occupy consecutive positions.
Complement method: A counting method in which invalid arrangements are subtracted from the total arrangements.
Factorial: The product of all positive integers from a number down to 1, written with an exclamation mark.
Clockwise: Movement around a circle in the same direction as the hands of a clock.
Anticlockwise: Movement around a circle opposite to the direction of clock hands.
Facing north: A direction in which a person's own left and right agree with the reader's left and right.
Facing south: A direction in which a person's own left and right are reversed compared with the reader's view.
Reference position: A fixed seat or person used to describe the relative positions of all other people.
Adjacent: Located next to another position without a gap.
Middle position: The central seat of a row when the number of seats is odd.

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Analytical and Logical Reasoning shortcuts

Choosing the stronger supporting argument

Choose the option that directly explains why the claim is true. Prefer relevant evidence with a clear link to the claim, not a statement that is merely related.

Identify the claim and ask, “Why would this make the claim true?”
Reject personal opinions, unrelated facts and extreme statements.
Example: “Teachers should be paid more” is supported by “They shape future generations.”

This shortcut does not apply when the option gives direct statistical or experimental evidence that is stronger than a general reason.

Finding the pattern in letter and number series

Check letters and numbers separately. Look for equal increases, alternating changes, multiplication, or a repeating pattern.

Convert letters to positions if needed, such as A = 1 and D = 4.
Find the change in the letters and in the numbers independently.
Example: A2, D4, G6, J8 has letters increasing by 3 and numbers by 2, so the answer is M10.

Do not assume a simple addition pattern if alternating or multiplication patterns fit all the given terms.

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