In how many ways can we arrange 3 boys and 3 girls in a circular table if:

 

1. 2 boys hate each other and would not want to sit beside each other
2. the 3 boys and 3 girls form 3 couples and they are to sit beside each other
3. b1 and b2 should sit opposite each other
4. no constraints
5. boys and girls are to alternate
6. all boys and girls are together

 

1. 72 ways  

2. 16 ways 

3. 48 ways 

4. 120 ways

5. 72 ways 

6. 36 ways.

Step-by-step explanation

Let the three boys be b1, b2, b3 and three girls be g1, g2, g3.

1. Total number of ways boys and girls sit in a circular table,

 (n-1)! = (6-1)! = 120

Consider 2 boys sitting beside each other, so we consider them as 1unit. Total number of persons is 5 .

Arrange 5 persons in circular way considering 2 boys as 1 unit and they can change their side so multiply with 2

(5-1)! *2 = 48

Number of ways that 2 boys would not sit beside each other = 120 - 48 =72

72 ways, 2 boys hate each other and would not want to sit beside each other

2. Let there be 3 boys and 3 girls form 3 couples, so we consider them as 1 unit each so 3 units for 3 couples and they can change their sit each of 2 ways.

n = (3-1)! * 2*2*2

n = 16

16 ways, the 3 boys and 3 girls form 3 couples and they are to sit beside each other.

3. Consider 2 boys sitting opposite to  each other in 2 ways.

Remaining can be arranged in (4)!

Total number of ways that b1 and b2 sit opposite to each other = 2 * 4! = 48

48 ways, b1 and b2 should sit opposite to each other. 

4. Total number of ways boys and girls sit in a circular table,

m = total persons sit in a circular table.

n = (m-1)!

n = (6-1)! 

n = 120

5. Let there be 3 seats for boys and 3 seats for girls. They sit alternately, no boys and no girl seated beside each other. Multiply with 2, if we change the seats of boys and girls.

n = 2*3!*3!

n = 72

72 ways, boys and girls are to alternate

6. Let all boys and girls sit together. 

n = 3!*3!

n = 36

36 ways, all boys and girls will sit  together