(a) Get how many arrangements have any two of A, B or C sitting together and don't worry about whether the third is with them or not.
(b) Get how many arrangements have all three of A, B and C sitting together.
Then subtract (b) from (a) ....
(a) = (number of ways of choosing a 'unit' of two from the three) times (number of ways of arranging within the 'unit') times (number of ways of arranging the 'unit' and the remaining six people) =
(b) = (number of ways of arranging within a 'unit' of three) times (number of ways of arranging the 'unit' of three and the remaining five people) =
So I get
At the moment I can't really see where the third term comes from .... someone better at arranging than I am might jump in and explain .....