Seating Arrangement - 1 - Free MCQ Practice Test with solutions, UPSC

Analyzing Each Arrangement:

Arrangement I: B P Q O K L R A Z M N

• Condition 1: Blue shirts (B) are at positions 1 and 8. They are not consecutive.(Right)
• Condition 2: Red shirts (R) are at positions 6, 7, and 10. Three reds are together (positions 6, 7, 10). (Right)
• Condition 3: Blue (B) at position 1 and Green (Q) at position 3 are not consecutive. (Right)
• Condition 4: Green (Q) is at position 3, not 2 or 9. (Right)
• Condition 5: Red (R) is at position 10, which is an extreme end. 

Since Condition 5 is violated, Arrangement I is not consistent.

Arrangement II: A P L K Z M O Q B R N

• Condition 1: Blue shirts (B) are at positions 8 and 9. They are consecutive. 

Since Condition 1 is violated, Arrangement II is not consistent.

Arrangement III: A N Q Z M K L P B R A

• Condition 1: Blue shirts (B) are at positions 8 and 10. They are not consecutive. (Right)
• Condition 2: Red shirts (R) are at positions 9 and 10. Only two reds are together, not three. 

Since Condition 2 is violated, Arrangement III is not consistent.

Arrangement IV: N M B Q R P L K Z A

• Condition 1: Blue shirts (B) are at positions 3 and 8. They are not consecutive. 
• Condition 2: Red shirts (R) are at positions 4, 5, and 6. Three reds are together. (Right)
• Condition 3: Blue (B) at position 3 and Green (Q) at position 4 are consecutive. (Wrong)

Since Condition 3 is violated, Arrangement IV is not consistent.

Arrangement V: B Z Q R L K M A P N

• Condition 1: Blue shirts (B) are at positions 1 and 10. They are not consecutive. (Right)
• Condition 2: Red shirts (R) are at positions 4, 5, and 6. Three reds are together. (Right)
• Condition 3: Blue (B) at position 1 and Green (Q) at position 3 are not consecutive. (Right)
• Condition 4: Green (Q) is at position 3, not 2 or 9. (Right)
• Condition 5: Red (R) is not at extreme ends. (Right)

All conditions are satisfied. (Right)

Arrangement VI: A R M Z K L P Q N B

• Condition 1: Blue shirts (B) are at positions 10. There is only one blue shirt. (Right)
• Condition 2: Red shirts (R) are at positions 2, 3, and 4. Three reds are together. (Right)
• Condition 3: Blue (B) at position 10 and Green (Q) at position 8 are not consecutive. (Right)
• Condition 4: Green (Q) is at position 8, not 2 or 9. (Right)
• Condition 5: Red (R) is at position 2, which is not an extreme end. (Right)

All conditions are satisfied. (Right)

Let the people who wear a blue, red and green shirt be denoted by b, r and g respectively. Restrictions on the seating arrangement:
1. Two b’s must not be together.
2. Three r’s must be together.
3. A ‘b’ and a ‘g’ must not be together.
4. A ‘g’ cannot sit on a chair numbered 2 or 9.

Case I: A person wearing a green shirt is sitting on chair numbered 1. It is only possible if another person wearing a green shirt sits on chair numbered 2, but this violates restriction number 4. Hence, this is also not possible.

Case II: A person wearing a blue shirt sits on chair numbered 1. The six seating arrangements that are possible are as follows.

Now, we see that the cases 4, 5 and 6 are just obtained by reversing the cases 1, 2 and 3 respectively. It can be concluded that in any possible seating arrangement, the chairs numbered 1 and 10 are always occupied by people wearing blue shirts. It is given that the number of people wearing a blue shirt is 3. Looking at the table given in the question, we observe that in each of the six arrangements two out of the three different people i.e. A, B and N always sit on chairs numbered 1 and 10.
Hence it can be concluded that the people who wear a blue shirt are A, B and N From the given table the person wearing a blue shirt can never sit on chairs numbered 2, 4, 7 and 9. So, (in arrangement I), A, B and N sitting on chairs numbered 1, 7 and 10 is inconsistent. Also, the people wearing red shirts sit on chairs numbered 2 and 9 and in all the possible arrangements five different people namely P, Q, M, Z and R are sitting on chairs numbered either 2 or 9. Therefore, P, Q, M, Z and R are wearing red shirts and K and L are wearing green shirts. A, B and N are wearing blue shirts. Hence, N is the answer.

Let the people who wear a blue, red and green shirt be denoted by b, r and g respectively. Restrictions on the seating arrangement:
1. Two b’s must not be together.
2. Three r’s must be together.
3. A ‘b’ and a ‘g’ must not be together.
4. A ‘g’ cannot sit on a chair numbered 2 or 9.

Case I: A person wearing a green shirt is sitting on chair numbered 1. It is only possible if another person wearing a green shirt sits on chair numbered 2, but this violates restriction number 4. Hence, this is also not possible.

Case II: A person wearing a blue shirt sits on chair numbered 1. The six seating arrangements that are possible are as follows.

Now, we see that the cases 4, 5 and 6 are just obtained by reversing the cases 1, 2 and 3 respectively. It can be concluded that in any possible seating arrangement, the chairs numbered 1 and 10 are always occupied by people wearing blue shirts. It is given that the number of people wearing a blue shirt is 3. Looking at the table given in the question, we observe that in each of the six arrangements two out of the three different people i.e. A, B and N always sit on chairs numbered 1 and 10.
Hence it can be concluded that the people who wear a blue shirt are A, B and N From the given table the person wearing a blue shirt can never sit on chairs numbered 2, 4, 7 and 9. So, (in arrangement I), A, B and N sitting on chairs numbered 1, 7 and 10 is inconsistent. Also, the people wearing red shirts sit on chairs numbered 2 and 9 and in all the possible arrangements five different people namely P, Q, M, Z and R are sitting on chairs numbered either 2 or 9. Therefore, P, Q, M, Z and R are wearing red shirts and K and L are wearing green shirts. K and L are wearing green shirts. Hence, K is the answer.

Let the people who wear a blue, red and green shirt be denoted by b, r and g respectively. Restrictions on the seating arrangement:
1. Two b’s must not be together.
2. Three r’s must be together.
3. A ‘b’ and a ‘g’ must not be together.
4. A ‘g’ cannot sit on a chair numbered 2 or 9.

Case I: A person wearing a green shirt is sitting on chair numbered 1. It is only possible if another person wearing a green shirt sits on chair numbered 2, but this violates restriction number 4. Hence, this is also not possible.

Case II: A person wearing a blue shirt sits on chair numbered 1. The six seating arrangements that are possible are as follows.

Now, we see that the cases 4, 5 and 6 are just obtained by reversing the cases 1, 2 and 3 respectively. It can be concluded that in any possible seating arrangement, the chairs numbered 1 and 10 are always occupied by people wearing blue shirts. It is given that the number of people wearing a blue shirt is 3. Looking at the table given in the question, we observe that in each of the six arrangements two out of the three different people i.e. A, B and N always sit on chairs numbered 1 and 10.
Hence it can be concluded that the people who wear a blue shirt are A, B and N From the given table the person wearing a blue shirt can never sit on chairs numbered 2, 4, 7 and 9. So, (in arrangement I), A, B and N sitting on chairs numbered 1, 7 and 10 is inconsistent. Also, the people wearing red shirts sit on chairs numbered 2 and 9 and in all the possible arrangements five different people namely P, Q, M, Z and R are sitting on chairs numbered either 2 or 9. Therefore, P, Q, M, Z and R are wearing red shirts and K and L are wearing green shirts.

Option (1): A (Blue), P (Red), R (Red) and L (Green): Permissible
Option (2): N (Blue), Q (Red), K (Green) and Z (Red): Permissible
Option (3): K (Green), A (Blue), N (Blue) and Z (Red): Not Permissible
Option (4): B (Blue), L (Green), M (Red) and Q (Red): Permissible.

As we need to use only two colours, in any row or column these two coloured beads will be
placed alternately like

So we cannot place Red coloured beads at position 1 or two as between any two Red beads there must at least two beads (at least one green and at least one Blue). Hence, we can use only Green and Blue coloured beads.

We can have two possible configurations:

Configuration 1: Green bead is placed at top left corner

Configuration 2: Blue bead is placed at top left corner

Between Any two Red beads there must be at least two Beads. So any Row or column there can be maximum two red beads. If we place two red beads in each row then two columns will have three red bead which cannot be accepted.

The above configuration is not correct.

So in the third row we will place only one Red bead at the middle of the third row. Also we will adjust other rows so that between any two Red beads there are at least two beads in any column.

So maximum 9 Red beads are possible in any configuration. At remaining places Green and Blue coloured beads can be placed in such way that all the conditions given are satisfied. There are multiple configurations are possible. One of the configurations is given as below.

To minimise number of Blue beads we need to maximise number of Red and Green beads. From the previous question solution, Maximum no. Red beads can be 9. The row in which has two red beads, we will place two green and one Blue bead additionally.

The row with only one red bead we will place two green and two blue beads additionally. So overall there will be minimum 6 Blue beads.

We can place maximum 6 more beads as shown below.