What I'm thinking: Find total ways that the ten people can be seated, which is 10!.
Then I take that number and subtract the ways the these two people would be seated next to each other. I do this by treating these two people as a single space, which leaves the eight other students plus that space consisting of the two. This would mean 9!
Then, 10! – 9! = 3265920 ways for the ten people to be seated so that a certain to are not next to each other.
Best Answer
Another way of thinking is. You can freely position 9 people (including one of the special two), therefore $9!$. However the remaining one cannot sit next to the other special person, so only 8 possible positions. Overall $8 \times 9!$