Seven friends sit down at a round table for dinner. Two of them, Faiz and Ishan, have quarrelled and must not be given neighbouring seats. In how many ways can the seven be seated?

Round a table only the relative order counts, so seven people can be seated in (7 − 1)! = 6! = 720 ways. To count the seatings where the quarrelling pair ARE neighbours, treat them as one unit: six units round the table give (6 − 1)! = 120 seatings, and the pair can swap seats within the unit in 2 ways, so 240. Subtract: 720 − 240 = 480. Trap 240 answers 'must sit together'. Trap 720 ignores the quarrel. Trap 5040 = 7! is the straight-row count and forgets that rotating the whole table gives the same seating.