Let the people be A,B,C,D,E,F
Suppose A and B insist on sitting next to each other
Then we have
these 5 things to arrange in a line:
AB,C,D,E,F
which is 5! ways,
plus
these 5 things to arrange in a line:
BA,C,D,E,F
which is also 5! ways.
Answer 2×5! = 2×120 = 240 ways.
Edwin