6 friends went to the cinema to watch a new movie. Then they realized that 2 of them are a couple and cannot be separated. How many different ways they can sit in the movie theatre?
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To keep the couple together, we temporarily treat them as one person. Then we are arranging five "people", which can be done in 5 ! ways. Once that's done, the couple itself can switch seats among themselves, which can be done in 2 ! ways. So the number of ways the six can sit keeping the couple together is 5 ! ⋅ 2 ! = 2 4 0