There are 3 adults and 6 children lining up, and the adults don't want to stand next to each other. How many ways are there to line up?
There are some basic positions for the queue, with A representing an adult and C representing a child.
ACACACCCC, ACACCACCC, ACACCCACC, ACACCCCAC, ACACCCCCA
ACCACACCC, ACCACCACC, ACCACCCAC, ACCACCCCA
ACCCACACC, ACCCACCAC, ACCCACCCA
ACCCCACAC, ACCCCACCA
ACCCCCACA
Looks like there are 5 + 4 + 3 + 2 + 1 = 15 positions.
In each position, there are 3! ways to order the adults, and 6! ways to order the children.
Therefore, there are \(15 \cdot 3! \cdot 6! = \fbox{64800} \text{ ways}\) :D