In how many different ways can 6 girls and 6 boys sit if boys don't want to sit next to each other?

Case 1 : The boys and girls are arranged in a row.

The arrangement should be : _G_G_G_G_G_G_
where each G stands for a girl, and each boy takes one of the sit labeled as "_".

Ways of arrangement of the 6 girls = P(6,6) = 6!
Ways of arrangement of the 6 boys to the 7 seats = P(7, 6) = 7!/(7-6)! = 7!

Total ways of arrangement = 6! × 7! = 3,628,800


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Case 2 : The boys and girls are arranged in a circle.

Ways of arrangement of the 6 girls = (6 - 1)! = 5!
Each boy takes a seat between 2 girls.
Ways of arrangement of the 6 boys = P(6,6) = 6!

Total ways of arrangement = 5! × 6! = 86400