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Linear Arrangement

Example-1
   Directions:
  1. Five friends, A, B, C, D and E, are sitting on a bench in a park.
  2. A is sitting next to B.
  3. C is sitting next to D.
  4. D is not sitting with E.
  5. A is to the right of B and E.
  6. E is at the left end of the bench.
  7. C is at the second position from the right.
  8. A and C are sitting together.

 
  1. Where is A sitting?
    1. Between B and D
    2. Between D and C
    3. Between C and E
    4. Between E and D
    5. Between B and C
  2. Who are sitting on either side of C?
    1. A and E
    2. A and D
    3. B and D
    4. D and E
    5. A and B
  3. What is the position of B?
    1. Extreme left
    2. Centre
    3. Second from left
    4. Second from right
    5. None of these
  4. Who is sitting at the centre?
    1. B
    2. D
    3. C
    4. E
    5. A
  5. What is the position of D?
    1. Extreme right
    2. Second from left
    3. Third from left
    4. Extreme left
    5. None of these

Solution

In the given statements, observe that statements (vi) and (vii) constitute definite information. Statements (ii), (iii), (v) and (viii) constitute comparative information, whereas statement (iv) gives negative information.

 

Starting with the definite information given in statements (vi) and (vii), we can draw the following seating diagram:

 

E _ _ C _

 

Now, let us go to comparative information that tells us about E and C. Statements (viii) and (ii) state that A and C are sitting together and also A and B are next to each other. It implies that A is sitting between B and C. Further, statement (v) states that A is to the right of B and E.

 

This leads to the following position:

 

E B A C _

 

Now, consider statement (iii). C is sitting next to D. On plotting this information we get the complete seating diagram, as under:

 

E B A C D

 

Based on the above diagram, the questions can be answered easily.
 

Answers:

  1. (5)
  2. (2)
  3. (3)
  4. (5)
  5. (1)
 

Example-2

Directions:
Seven persons, Anil, Bharat, Chand, Dinesh, Eshwar, Fazal and Ganesh, who have gathered at a picnic spot, are sitting in a row facing a photographer.

  1. Chand and Dinesh are sitting next to each other.
  2. There are exactly four persons between Bharat and Ganesh.
  3. Dinesh is sitting to the immediate right of Bharat.

 
  1. If Anil and Eshwar are separated exactly by two persons, then who is sitting to the left of Ganesh?
    1. Anil
    2. Eshwar
    3. Fazal
    4. Chand
    5. None of these
  2. If Bharat is sitting at one end, then who is sitting at the other end of the row?
    1. Anil
    2. Eshwar
    3. Ganesh
    4. Cannot be determined
    5. None of these
  3. If Bharat is not sitting at either end of the row, then who among the following has as many persons on his left as on his right?
    1. Chand
    2. Fazal
    3. Eshwar
    4. Ganesh
    5. None of these
  4. If Eshwar is sitting to the right of Bharat and Anil is separated from Eshwar by exactly three persons, then who is sitting to the immediate left of Ganesh?
    1. Fazal
    2. Anil
    3. Eshwar
    4. Chand
    5. None of these
  5. In how many different ways can the seven persons sit in a row, without violating the given conditions?
    1. 3
    2. 10
    3. 2
    4. 12
    5. None of these

Solution

First, let us represent the given conditions by noting down the position of each person.

 

Chand and Dinesh are sitting next to each other (statement no. i). It can be represented as:

 

C D or D C

 

There are exactly 4 persons between Bharat and Ganesh (Statement no. ii). This can be represented as under:

 

B _ _ _ _ G or G _ _ _ _ B

 

Dinesh sits to the immediate right of Bharat. This can be represented as B D.

 

The next step is numbering the seats from 1 to 7 (since there are seven persons).

 

Since Dinesh is to the right of Bharat, and Chand and Dinesh sit next to each other, Chand can only sit to the immediate right of Dinesh. Thus, Bharat, Dinesh and Chand will sit in the order B, D, and C. Since there are four persons between Bharat and Ganesh, Bharat can be positioned in seats 1, 2, 6 or 7. However, if Bharat is in seat 6 or 7, then Chand and Dinesh cannot sit to the right of Bharat. Hence, Bharat can be positioned only in two seats either 1 or 2. Accordingly, we get two arrangements.

Arrangement I:

1

2

3

4

5

6

7

Bharat

Dinesh

Chand

 

 

Ganesh

 


Arrangement I:

1

2

3

4

5

6

7

 

Bharat

Dinesh

Chand

 

 

Ganesh


These are the only two arrangements possible for the four persons—Ganesh, Chand, Dinesh and Bharat.

 

The other three persons Anil, Eshwar and Fazal can be seated in the three vacant slots in any order as no specific information is given about their positions.
 

Now, let us solve the questions.

  1. Anil and Eshwar are separated exactly by two persons.
     
    Arrangement I is the only possible option; because, in arrangement II, Anil and Eshwar cannot have exactly two persons between them. Hence, the arrangement can be as shown below:
     
    1 2 3 4 5 6 7
    Bharat Dinesh Chand Anil/Eshwar Fazal Ganesh Eshwar/Anil

    So, Fazal sits to the immediate left of Ganesh.
     
    Answer: (3)
  2. Bharat is sitting at one end. This implies that they are sitting as per arrangement I. In this arrangement, any one among the three persons Anil, Eshwar or Fazal can be positioned in seat no. 7, i.e., another end. Hence, we cannot determine as to who is sitting at the other end of the row.
     
    Answer: (4)
  3. If Bharat is not at either end, then only arrangement II is possible. The person who has as many persons to his left as to his right is the one who sits in the middle seat, i.e., seat no. 4 (since there are seven persons). Chand is sitting in seat no. 4.
     
    Answer: (1)
  4. If Eshwar and Anil are separated exactly by three persons, then only arrangement II is possible. Eshwar can sit in seat no. 1 or 5 and Anil can sit in seat no. 5 or 1. Since it is stated that Eshwar sits to the right of Bharat, Eshwar has to be in seat no. 5. Hence, the arrangement is as given below:
     
    1 2 3 4 5 6 7
    Anil Bharat Dinesh Chand Eshwar Fazal Ganesh
     

    Answer: (1)
     
    The person sitting to the immediate left of Ganesh is Fazal.
  5. As we know, there are two possible arrangements—I and II. In each arrangement, the remaining three people Anil, Eshwar and Fazal, can sit in the remaining 3 seats in six different ways. Thus, these seven people can be seated in 12 different ways.
     
    Answer: (4)

 

Example-3

Directions:

Seven girls, Ramya, Simran, Vinutha, Mandakini, Urmila, Preethi and Kinnera are sitting in a row. Simran is sitting to the immediate left of Vinutha and third to the right of Ramya, whereas Mandakini, who is sitting at the extreme left, is sitting next to Kinnera.

  1. Who is sitting to the immediate left of Ramya?
    1. Mandakini
    2. Urmila
    3. Kinnera
    4. Preethi
    5. Cannot be determined
  2. If Vinutha and Kinnera mutually exchange their places without any change in the arrangement of other positions, who will be sitting to the immediate left of Ramya?
    1. Kinnera
    2. Preethi
    3. Urmila
    4. Vinutha
    5. None of these
  3. If only Simran is sitting between Preethi and Vinutha, who is sitting exactly at the centre?
    1. Preethi
    2. Urmila
    3. Vinutha
    4. Ramya
    5. None of these
  4. If Ramya exchanges her place with Mandakini and Vinutha exchanges her place with Urmila, then how many girls will be there between Urmila and Ramya?
    1. Two
    2. Three
    3. Four
    4. Five
    5. Cannot be determined
  5. In addition to the above statements, which of the following statements cannot specify the seating position of every girl?
    1. Ramya and Simran have two girls sitting between them.
    2. Vinutha and Urmila have two girls between them.
    3. Mandakini and Preethi have two girls between them.
    4. There are two girls sitting between Preethi and Kinnera.
    5. Every statement given above specifies the seating arrangement of every girl.

Solution

Based on the data given above, let us draw the seating diagram. Let us represent these seven girls by the first letter of their name as R–Ramya, S–Simran, V–Vinutha, M–Mandakini, U–Urmila, P–Preethi and K–Kinnera.

 

M–Mandakini, who is sitting at the extreme left, is sitting next to Kinnera.

 

M K _ _ _ _ _

 

L R

 

Simran is sitting to the immediate left of Vinutha and third to the right of Ramya. This can be denoted as under:

 

R _ _ S V

 

Combining the above two diagrams, we can deduce that Vinutha is sitting at the extreme right end of the row.

 

Thus, the seating arrangement will be:

 

M K R _ _ S V

 

The two vacant positions between Ramya and Simran are to be occupied by Preethi and Urmila.

  1. From the final seating diagram, we notice that Kinnera is sitting to the immediate left of Ramya.
     
    Answer: (3)
  2. If Vinutha and Kinnera mutually exchange their positions, then Vinutha will be sitting to the immediate left of Ramya.
     
    Answer: (4)
  3. If Simran sits between Preethi and Vinutha, then the seating diagram takes the following form:
     
    M K R U P S V
     
    It is clear that Urmila is sitting at the centre of the row.
     
    Answer: (2)
  4. The seating arrangement is:
     
    M K R P/U U/P S V
     
    If Ramya exchanges her place with Mandakini, and Urmila with Vinutha, then the seating arrangement will be as given below:
     
    R K M P/V V/P S U
     
    Observe from the above seating diagram that there are five girls sitting between Ramya and Urmila.
     
    Answer: (4)
  5. Based on statement no. (1), the seating arrangement will be:
     
    M K R P/U U/P S V
     
    The position of all the seven girls cannot be determined since the position of Preethi and Urmila are not certain.
     
    Based on statement no. (2), the seating arrangement is as shown below:
     
    M K R U P S V
     
    Based on statement no. (3), the seating arrangement is as shown below:
     
    M K R P U S V
     
    Based on statement no. (4), the seating arrangement is as shown below:
     
    M K R U P S V
     
    From the above diagrams, we can infer that only statement no. (1) does not specify the seating position of every girl.
     
    Answer: (1)
 

Example-4

Directions:
Six persons A, B, C, D, E and F are sitting in two rows, three in each, facing each other.

  1. E is not at the end of any row.
  2. D is second to the left of F.
  3. C, the neighbour of E, is sitting diagonally opposite to D.
  4. B is the neighbour of F.

 
  1. Who is facing B?
    1. C
    2. A
    3. D
    4. F
    5. E
  2. Who among the following are in one of the two rows?
    1. C, E, B
    2. D, B, F
    3. A, B, F
    4. A, E, F
    5. F, B, C
  3. Who among the following are sitting diagonally opposite to each other?
    1. A and C
    2. F and C
    3. A and F
    4. A and B
    5. D and A
  4. Who among the following are sitting in the same row?
    1. A and B
    2. C and F
    3. C and B
    4. E and D
    5. A and E
  5. If D and E exchange their seats, who will be the neighbours of D in the new seating arrangement?
    1. Only C
    2. Only B
    3. E and B
    4. C and A
    5. Only A

Solution

In this example, all the six people are not facing the same direction. If three people who are sitting in one row are facing north, the remaining three people who are sitting in the opposite row face south.

 

In such problems, we should be very cautious in considering left and right directions. The left direction of one row becomes the right direction for the other row.

 

Again, in this problem, it appears as if we do not have any definite information, as the given information is either comparative or negative information. However, on correct interpretation of the statements (i) and (ii) we arrive at definite information, which is hidden.

 

In terms of statement (i), E is not at the end of any row. This implies that E is in the middle of a row, since there are only three persons in each row. Statement (ii) states that D is second to the left of F. This implies that D is at the left end of a row, and F is at the right end of the same row. This is because there are only three persons in a row.

 

According to statement (iii), C is diagonally opposite to D and is a neighbour of E. On representing all these information in the seating diagram, it appears as under:

 

  Left
_ E C
D _ F
Left  


As per statement (iv), B is the neighbour of F. This implies that B is in the middle of the row and is sitting between D and F. The only remaining person is A. He/she should sit in the vacant place beside E.

 

The complete seating diagram is as shown below:
 

  Left
A E C
D B F
← Left  


Now we can answer the questions.

 

Answers:

  1. (5)
  2. (2)
  3. (3)
  4. (5)
  5. (4)




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