### Circular Permutations ~ 3 Boys & 3 Girls

In how many ways can 3 boys and 3 girls be seated at a round table such that exactly 2 boys sit together?

Solution

Circular permutations without any restriction = (6 - 1)! = 120

Circular permutations when no 2 boys are together which is B G B G B G situation is a circle = 2! $\times$ 3! = 12

Circular permutations when all the 3 boys are together = (4 - 1)! $\times$ 3! = 36

Circular permutations for exactly 2 boys sitting together =  120 - (12 + 36) = 72