Five boys and four girls of a dance troupe are to stand in a straight line for a photograph so that boys and girls come alternately. In how many ways can the line be formed?

With 5 boys and 4 girls the only alternating pattern possible is B G B G B G B G B — the line must both start and end with a boy, since a girl-first line would need as many girls as boys. So the 5 boys fill the five boy-slots in 5! = 120 ways and the 4 girls fill the four girl-slots in 4! = 24 ways: $120\times 24=2880$. Trap 5760 doubles the count for a girl-first line that cannot exist here. Trap 362880 = 9! ignores the alternating rule. Trap 1440 halves the correct count for a symmetry the line does not have.