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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?

 May 2, 2020
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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

 May 2, 2020

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