Arrangements

Linear Arrangements 1

Linear Arrangements 1

Arrangements involve placing objects or persons in a particular sequence or order. Problems on this type can be encountered very frequently in any aptitude exam. In this type of problems, information is provided about the positions of certain objects or persons. Using this information and applying logical deduction one is supposed to figure out the entire sequence of objects/persons. Arrangements are of two types:

1. Linear Arrangement
2. Circular Arrangement

When we talk of Linear Arrangements then it implies placing of objects/persons along a straight line or a row. So there is a starting point and a terminating point in a row. The problem set may involve even multiple rows. As the objects/persons are arranged along a row, the position of these objects/persons assumes great importance. Before moving to the problem sets, let’s first understand some of the terminologies that will be quite helpful.

Example 1: Seven friends Mowgli, Bagheera, Baloo, Kaa, Raksha, Shere Khan and Hathi went together for the first day show of the movie ‘The Jungle Book’ and were sitting in a row facing the screen. Baloo and Kaa were sitting next to each other. There were exactly four persons between Bagheera and Hathi. Kaa was sitting to the immediate right of Bagheera.

If Mowgli and Raksha were separated exactly by two persons, then who sits to the immediate left of Hathi?

A. Mowgli                        B. Raksha                  C. Shere Khan             D. Baloo

If Bagheera was not sitting at either extreme of the row, then who among the following had as many persons on his left as on his right?

A. Kaa                 B. Shere Khan                    C. Baloo             D. Hathi

If Bagheera sat at one extreme, then who was at the other extreme?

A. Mowgli                 B. Raksha                       C. Hathi              D. Cannot be determined

If Raksha sat to the right of Bagheera, and Mowgli was separated from Raksha by exactly three persons. Then, who was sitting to the immediate left of Hathi?

A.Shere Khan           B. Mowgli                      C. Raksha            D. Baloo

In how many different ways could have the seven persons sit in a row?

A.3                            B. 2                               C. 10                     D. 12

Solution: As there were exactly four persons between Bagheera and Hathi, there are four different arrangements possible:

Bagheera  _  _  _  _  Hathi  _      (1)

Hathi  _  _  _  _  Bagheera  _     (2)

_  Bagheera  _  _  _  _  Hathi     (3)

_  Hathi  _  _  _  _  Bagheera     (4)

Now as Kaa was sitting to the immediate right of Bagheera, so (4) gets eliminated and so it becomes:

Bagheera  Kaa  _  _  _  Hathi  _      (1)

Hathi  _  _  _  _  Bagheera  Kaa     (2)

_  Bagheera  Kaa  _  _  _  Hathi     (3)

But also Baloo and Kaa were sitting next to each other. So, case (2) gets ruled out and we get :

Bagheera   Kaa   Baloo   _   _    Hathi   _      (1)

_   Bagheera  Kaa   Baloo  _   _   Hathi         (3)

1. Given that Mowgli and Raksha were separated by exactly two persons, so case (1) remains and the final arrangement becomes:

Bagheera     Kaa     Baloo    Mowgli/Raksha   Shere Khan  Hathi    Raksha/Mowgli

Thus, Shere Khan sits to the immediate left of Hathi. So, option (C) is correct.

1. Given that Bagheera was not sitting at either extreme of the row, so case (3) remains

_   Bagheera   Kaa  Baloo   _    _   Hathi       (3)

Thus, Baloo has number of persons on his left as on his right. Thus, option (C) is correct.

1. Given that Bagheera sat at one extreme, so case (1) remains

Bagheera   Kaa   Baloo   _   _    Hathi   _

Now the person on the other extreme could be Mowgli, Raksha or Shere Khan. So option (D) is correct.

1. Given that Raksha sat to the right of Bagheera, and Mowgli was separated from Raksha by exactly three persons, the final arrangement becomes:

Mowgli   Bagheera  Kaa   Baloo  Raksha   Shere Khan   Hathi       (3)

Thus, Shere Khan was to the immediate left of Hathi. So, option (A) is correct.

1. There are two cases (1) and (3). Now if we consider case (1) i.e.

Bagheera   Kaa   Baloo   _   _    Hathi   _      (1)

There are three places and we have have three persons who can be placed in these places in 3! i.e. 6 ways.

Similarly for case (3), we can again have 6 ways of arranging them.

Thus, a total of 12 ways are there to arrange these seven friends. So, option (D) is correct.

Example 2: Eight friends Akash, Bhavika, Chameli, Dhruv, Ema, Farooq, Gayatri and Honey are standing from left to right not necessarily in the same order. The following information is available about them :

A. Gayatri is standing to the immediate right of Akash while Chameli is two places to the left of Akash

B. Bhavika is at the extreme left while Honey is standing exactly between Bhavika and Chameli

C. Farooq is standing somewhere to the left of Chameli while three persons are standing between Farooq and Dhruv

Who is standing at the third position from left?

A. Farooq                       B. Ema              C. Honey              D. Cannot be determined

Which of the following is true?

A.Farooq is standing next to Ema.

B. Dhruv is standing three places to the left of Gayatri.

C. Akash is sitting exactly between Dhruv and Gayatri

D. None of the statements is true.

Solution: Let’s call these persons by A,B,C,D,E,F,G and H.

From statement B we get three possible cases:

B  H  C  _   _   _   _  _     (1)   or      B _ H _ C _  _  _   (2)    or          B  _  _  H  _  _  C _  (3)

From statement C, F is standing to the left of C, thus, case (1) gets eliminated and if we combine it with statement A, then case (3) gets eliminated and we get

B _ H _ C _  A  G   (2)

Also three persons are between F and D and as F is to the left of C, we get the final arrangement as:

B    F    H    E    C    D    A    G    (From left to right)

1. Honey is standing at the third position form left. So, option (C) is correct.
2. Akash is sitting exactly between Dhruv and Gayatri. So, option (C) is correct.

Example 3: Seven chess pieces – 2 rooks, 2 knights, 2 bishops and a queen are placed next two each other not necessarily in the same order. The following information is available about their arrangement:

A. The two bishops are placed next to each other while a rook is on the extreme right.

B. The queen is not next to a knight and is not at any extreme.

C. A rook is exactly between a bishop and the other rook while no two rooks or the two knights are placed together.

Which chess piece is placed at the second position from the left?

A. Bishop         B. Rook             C. Knight               D. Queen

Which chess piece is to the immediate right of the queen?

A. Rook            B. Knight          C. Bishop               D. Cannot be determined

Which chess piece is placed at the sixth position from the left?

A. Queen         B. Rook            C. Knight                D. Cannot be determined

Solution: From statement A, rook is placed at the extreme right. So,

_  _  _  _  _  _  _  R

From statement C, a rook is placed exactly between a bishop and the rook. This gives the following possibilities:

_   _  _  _ B R R (1)   or  _  _  B  _  R  _  R   (2)  or  B  _  _  R  _  _  R   (3)

But as two rooks cannot be together (from C), case (1) gets ruled out.

Also as the two bishops are together, if we look at case (3), then

B  B  _  R  _  _ R . Now the queen cannot be next to a knight and also the two knights cannot be together. So, we cannot place these pieces if we consider this case. Thus, only case (2) remains.

Now there are two possibilities

_  B  B  _ R _ R  (2a)  or  _  _  B  B  R  _  R  (2b)

As Queen cannot be placed at any extreme position, So (2a) gives

K  B  B  Q/K  R   K/Q  R

And (2b) again can be ruled out as we won’t be able to place the queen and the two knights in the blank spaces.

Thus, the final arrangement becomes

Knight  Bishop  Bishop  Queen/Knight  Rook  Knight/Queen  Rook

1. Bishop is placed second from the left. So, option (A) is correct.
2. In any case Rook is to immediate right of the Queen. So, option (A) is correct.
3. Either the Queen or the Knight is placed at the sixth position from the left. So, option (D) is correct.

Many a time a question set on linear arrangements would involve more than one variable. For example let’s say there are seven friends standing in a row from left to right and they work in seven different multinational companies. So, in such situations we have to figure out the respective companies where they work and also their linear arrangement.

Example 4: Six ninjas- Susake, Kakashi, Jiraya, Tsunade, Itachi and Naruto are high level shinobis and each of them is a different style user from among- fire, earth, wind, wood, lightening and water not necessarily in the same order. They are standing in a queue from left to right and the following information is available about them:

A. The water style ninja is to the immediate left of Jiraya who himself is a wood style

B. Kakashi is at the extreme left while Itachi and the earth style user are next to each other

C. The fire style user is two places to the left of the wood style user

D. Tsunade who is not a lightening style user is just next to Naruto who is an earth style user

E. Sasuke is four places to the left of Tsunade while Itachi is next to the wood style user

Who is a fire style user?

A. Tsunade       B. Itachi          C. Kakashi            D. Cannot be determined

Which style user is to the immediate left of Itachi?

A.Water            B. Fire            C. Wind                 D. Wood

Which of the following is true?

A. Kakashi is a wind style user.

B. The wood style user is to the immediate left of the lightning style user.

C. Tsunade is a lightning style user sitting next to Jiraya.

D. Naruto is sitting two places to the right of the water style user.

Solution : From statement (1)  and (6) we get the water style is next to Jiraya while  Jiraya and Itachi are next to each other as Jiraya is a wood style user. Also from (2) and (4) we get Itachi is next to Naruto as he is an earth style user. We get :

From (4) we get Tsunade is next to Naruto while the fire style user is two places to the left of the wood style user i.e. Jiraya.

From (2) we get Kakashi is at the extreme left and so Sasuke is the water style user. Also from (4) we get Tsunade is not a lightning style user. Thus, Tsunade is a wind style user and Itachi is a lightning style user. The final arrangement becomes:

1. Clearly Kakashi is a fire style user. So, option (C) is correct.
2. Wood style user is to the left of Itachi. So, option (D) is correct.
3. Only statement B is correct

Sometimes the sets on linear arrangements involve more than one row. Surely as the number of rows increases, the level of difficulty of a set also gradually increases. Let’s take an example to understand such type of questions.

Example 5: Twelve houses are of different colours from among Blue, Black, White, Yellow, Orange, Green, Red, Maroon, Purple, Brown, Pink and Gray and are standing on two rows- I and II as shown in the figure. The 2 rows are facing each other:

Row 1:  ____     _____    _____       _____      _____     _____

Row 2:   ____     _____    _____      ______     ______    _____

The following information is known about them:

1. The blue house is to the left of the Maroon house and is in row I while the brown house is directly opposite to the Maroon house and is to the immediate right of the red house
2. Orange house is to the immediate right of the Pink house while there are exactly two houses between the blue and the pink house
3. There is no house to the left of the red house while the Purple house is next to the green house and is in row II
4. White house is in row I and is not directly opposite to the red house while the gray house is next to the green house
5. There are two houses which are directly opposite to each other and the colours of these houses begin with the same letter and there are no other houses whose name start with the same letter and are directly opposite to each other

Which color house is directly opposite to the White house?

A. Gray         B. Purple         C. Black           D.  Cannot be determined

Which colour house is three places to the left of the Maroon house?

A. White       B. Orange        C. Purple         D. cannot be determined

Which colour house is opposite to the green house?

A. Orange    B. White           C. Black          D. Cannot be determined

Which colour house is two places to the left of the Brown house?

A. Purple      B. Gray            C. Green        D. No house

Solution: If we consider statement (1), then we get:

From (3) we get Red house is at the extreme left in Row 2. The arrangement would look like:

From (2) we get Orange house is to the immediate right of the Pink house while there are exactly two houses between the Blue and the Pink house, So Pink house must be at the rightmost end of row 1 and Orange house just next to it. Also from (4), white house must not be opposite to the Red house, which further means it is next to the blue house.

From (3) and (4) we get, Purple, Green and Gray house must be together in row 2. This means either Black or yellow is the other house in that row.

From (5), there are only two combinations of houses that start with the same letter. i.e. Black and Blue and Purple and Pink ( Of the houses left to be placed)

Case1: Let the Black house be opposite to the Blue house. So, Purple cannot be opposite to the Pink house. The arrangement becomes:

Case2: Let the Purple house be opposite to the Pink house. So, Blue cannot be opposite to the Black house. The arrangement becomes:

1. Either the Gray or the Purple house is opposite to the White house. So, option (D) is correct.
2. Orange house is three places to the left of the Maroon house. So, option (B) is correct.
3. In both the cases, Orange house is opposite to the Green house. So, option (A) is correct.
4. No house is two places to the left of the Brown house. Option (D) is correct.

Example 6: Six buses B1, B2, B3 , B4, B5 and B6 operate between two stations Raipur and Durg. The following information is known about their departure and arrival on a particular day.

1. The last bus to leave Raipur was the first to reach Durg and it was not B3 or B6 while the first bus to leave Raipur was not the last to reach Durg.
2. B1 left earlier than B3 from Raipur and reached Durg later than B3 and B6.
3. B2 was the third bus to leave Raipur and reached Durg just after B4 but just before B3.
4. B4 was not the first bus to reach Durg.
5. There was only one bus leaving between B2 and B3 from Raipur and reached fifth at Durg.

Which was the first bus to leave Raipur?

A. B1         B. B3          C. B4             D. Cannot be determined

Which was the fifth bus to reach Durg?

A. B2         B. B6          C. B3             D. Cannot be determined

Which of the following is true?

A. B2 reached Durg before B5

B. The last bus to leave Raipur was B4.

C. B6 left Raipur before B3 and reached Durg after B3.

D. All the statements are true.

Solution: From statement (1), the last bus to leave Raipur was not B3 or B6. Also from (2), B1 left before B3, so B1 cannot be the bus that left Raipur the last. From (3) B2 was the third to leave Raipur and as B4 was not the first to reach Durg, this implies B4 was not the last to leave Raipur. Thus, B5 was the bust that was last to leave Raipur and it was the first to reach Durg.

 S. No Raipur Durg 1 B5 2 3 B2 4 5 6 B5

From statement (3) we get the order in which the buses reached Durg was: B4 > B2 > B3 and Also  B1 reached Durg after B3 and B6 . As the first bus to reach Durg was B5, so the correct order in which the buses reached Durg was:

B5  >  B4  >  B2  >  B3  >  B6  >  B1

 S. No Raipur Durg 1 B5 2 B4 3 B2 B2 4 B3 5 B6 6 B5 B1

From statement 5, the bus that left Raipur between B2 and B3 was the fifth to reach Durg. So that bus is B6 and also as B1 left before B3, so B3 cannot be the first bus to leave Raipur. This implies that B3 was the fifth bus to leave Raipur and B6 was the fourth bus to leave Raipur.

 S. No Raipur Durg 1 B5 2 B4 3 B2 B2 4 B6 B3 5 B3 B6 6 B5 B1

From statement (1), the first bus to leave Raipur, was not the last to reach Durg. So, B1 cannot be the first bus that left Raipur. So, final arrangement becomes:

 S. No Raipur Durg 1 B4 B5 2 B1 B4 3 B2 B2 4 B6 B3 5 B3 B6 6 B5 B1
1. B4 was the first bus to leave Raipur. Hence option (C) is correct.
2. B6 was the fifth bus to reach Durg. Hence option (B) is correct.
3. Only the third statement is true.