1. Introduction

An arrangement consists of a group of items (or people) one after the other. The order in which the items or people are arranged will be based on the details given in the question. For instance, we can have cars in a straight line in a road, people standing in a queue, people seated around a rectangular table, etc. These questions occur on CAT and most MBA entrance tests every year.

Arrangement is of two types – linear arrangement and closed group arrangement.

A linear arrangement is one where the items or people are stacked in a single line, one after the other. There is a beginning (first person or item) and an end (last person or item) of the line. For instance, people standing in a queue, animals in a row in a zoo, a set of items laid out in an increasing order of their cost, etc.

A closed group arrangement does not have a beginning or end, it will have a group of people seated or a group of items arranged in a circular manner. The most common arrangement is a circular one, but we will also have others, such as hexagonal, rectangular, etc. For instance, people can be seated in a circular or rectangular table, children sitting on a circular merry-go-round, a set of precious stones in a necklace arranged in a particular order. As you can infer, there is no rank or order to the items (first person, last people, etc.), just the arrangement.

We will have to infer where each person or item is placed in the arrangement, based on the information or clues given in the question. Let us solve a few questions to get a clear idea.

Concept Case 1

A, B, C, D, E, F and G are seated on 7 chairs arranged in a line. The following is known about them:

1) B is seated in the fourth chair from the left.
2) A is seated to the left of B.
3) C is seated to the immediate left of B.
4) D is seated 2 seats away from B. B is seated to the left of D.
5) There are 3 seats between E and F.

Solution The words “7 chairs arranged in a line” lets us know that this is a linear arrangement case.

Let us number the chair from left to right for identification purposes.

B is seated in the fourth chair and C is seated to the immediate left of B, hence, C sits in chair 3.
D is seated 2 seats from B, B is seated to the left of D. Thus D is seated two seats to the right of B

There are 3 seats between E and F. If E/F sits at seat 7 then F/E cannot be seated as the 3rd seat is occupied by C. So E/F and F/E have to occupy seats 1 and 5.

A is seated to the left of B, hence, the only seat A can take is seat 2.
The remaining seat, 7, is taken by the only remaining person, G.

This is the possible arrangement.

Concept Case 2

A, B, C, D, E and F are seated around a circular table in six equally spaced seats. The following information is known about them:

1) A, B and C do not sit opposite D, E and F respectively.
2) F is seated two seats away from D in the counter-clockwise direction.
3) A and E are seated opposite each other.
4) A is 4 seats away from F in the clockwise direction.

Solution:

The words “are seated around a circular table” lets us know that this is a circular arrangement case.

For identification purposes, let us number the seats in anticlockwise direction.

Let us seat A first – A is seated opposite to E. If A sits in seat 3, then E sits in seat 6.

F is 4 seats away from A in clockwise direction, and hence, F takes seat 5.
D is seated 2 seats away from F in anti/counterclockwise direction. Hence, D sits in seat 1.

The remaining two persons that need to be seated are B and C.
C cannot sit opposite to F, hence the only place C can sit is opposite to D at seat 4.
Hence, B takes the only remaining seat opposite to F, seat 2.

Concept Case 3

A, B, C, D, E, F, G and H are seated around a rectangle such that there is 1 person sitting along each of the shorter sides and there are three persons sitting along the each of the longer sides. The distance between any 2 adjacently seated persons is the same. The following information is known about them:

1) G is seated in the middle of one of the lengths and B is seated to the immediate left of G.
2) D is seated diagonally opposite B.
3) E, who is seated three seats away from A, is seated opposite F.
4) A is not seated opposite B.
5) There are two people seated between H and B.

Solution:

The words “are seated around a rectangle” lets us know that this is a closed arrangement case.

The shorter side of the rectangle is the breadth of the rectangle, the rectangle has two short sides, hence, there are a total of two people sitting at the shorter sides.

The longer sides have 3 people each, which is a total of 6 people.

Let us number the seats in anticlockwise direction for identification purposes.

G is seated at the middle of one of the lengths with B to the immediate left. Let G sit at 3 and so B sits at 2.
D is diagonally opposite to B, which isthe seat diagonally opposite to B is seat 6.

E is seated opposite to F. The only opposite seats available are on the shorter sides of the rectangle.
Hence, E and F are seated at 1 and 5 in no particular order.

A is not opposite to B, hence, A cannot sit at seat 8.
If E sits at seat 5 then A has to be seated 3 places to the right of E. The left side is already occupied by B. But this is not possible as A cannot be opposite to B.

Hence, E has to be seated at seat 1 and F at seat 5. Further, A is seated 3 places to the right of E at seat 4.
There are exactly 2 people seated between H and B. The only possible seat H can take is seat 7. Therefore, the remaining seat is taken by C who sits at seat 8.

Concept Case 4

A, B, C and D are sitting around a circular table such that two of them are facing the centre of the table, while two of them are facing away from the table (i.e., have their backs facing the table). The following is known about them:

1) A and C are seated opposite each other.
2) B is seated to the immediate left of C.
3) D faces the centre of the table.
4) A and B are either both facing the table or both not facing the table.
5) The space between any 2 chairs is the same.

Solution:

The words “are sitting around a circular table” lets us know that the case is a circular arrangement case. For identification purposes, let us number the seat in anticlockwise direction.

A and C are opposite to each other. Let A sit at 3, and so, C sits at 1.

B sits to the immediate left of C. Since we are not sure if C is facing inwards or outwards, let us solve for two different arrangements cases.

Case 1: C is facing inward

Case 2: C is facing outward

In both the cases, D faces the centre of the table. Hence D is inward facing.

In Case 1, C and D are facing the centre, and hence, A and B should face outwards, as 2 people face towards the the table and 2 people face away from the table.

In Case 2, C is facing outward and D is facing inward. We know that 2 people should face inward and 2 should face outward. This means that A and B either face inward and outward or outward and inward respectively.

This case is not permitted as A and B both should face towards the same directions, hence, we can eliminate this case.

Case 1 is the correct case and the correct arrangement.