Many ways to go
There are 10 people who are sitting in a round table. Two of them do not want to sit together. What are the number if ways in which they don't sit together?
The answer is 282240.
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1 solution
Total number of ways in which all 10 people can sit; (10-1)! = 9! Let's fix Bob and Bill's positions to be opposite each other. Rest 8 can sit in 8! ways. Now, BOB and Bill toggle their position; rest 8 will rearrange themselves in 8! way again.
Bob and Bill don't sit at the opposite tables = 9!-2x8! = 362880-2*40320=282240