Four men and four women are to be seated round a table so that no two men are neighbours. In how many ways can they be seated?

Seat the four women first: round a table that can be done in (4 − 1)! = 3! = 6 ways. Four women sitting round a table leave exactly four gaps between them, and no two men can be neighbours only if each gap takes exactly one man — the four men fill those gaps in 4! = 24 ways. Total = $6\times 24=144$. Trap 576 = 4! × 4! seats the women as if they were in a row and forgets that rotating the whole table repeats a seating. Trap 5040 = 7! is the unrestricted count for eight people at a table. Trap 288 halves 576 for a reflection that does not apply at a table.