There is no need for circular permutation formulas. You can pick the 3 award winners in 10C3 = (10)(9)(8)/3! = 120 ways. We can then discard the arrangements we don't want to count. There are 10 ways to pick three winners who are all in adjacent seats (because from any one of the ten people, picking the next two people clockwise around the table will produce a different selection of three adjacent people). Similarly, there are 10 ways to pick two people in adjacent seats, and when we pick two adjacent people, there will be 6 people left at the table who are not adjacent to either of those two, so there are 6*10 = 60 ways to pick two people in adjacent seats and a third in a non-adjacent seat. So there are 10 + 60 = 70 ways to pick three people from this table so that at least two are in adjacent seats, and 50 remaining ways to pick three people so that none are adjacent, and the answer is 50/120 = 5/12.

Answer: C