1. The problem statement, all variables and given/known data 6 children are sitting arounbd a round table, with each chair numbered from 1 to six. In how many ways can we rearange them so that no child sits across from the child who was across him before. 2. Relevant equations 3. The attempt at a solution Because the chairs are numbered it makes a difference where each child sits. To seat the 1st child, A, we have 6 options, and 5 open seats remain. To seat the child that previously sat across from A, A', we have 4 options. And 4 seats remain To seat the 3rd child, B, we have 4 options, and 3 open seats remain. To seat the child that previously sat across from B, B', we have 2 options. And 2 seats remain To seat the 5th child, C, we have 2 options, and 1 open seats remain. To seat the child that previously sat across from C,C', we have 1 option. Notice that by the way we sat the other children, C and C'can not be across from each other. in total there are 6*4*4*2*2= 384.