My solution is:
- Choose $1$ man from $6$ men: $6$ ways
- Choose $1$ woman from $6$ women: $6$ ways
- Arrange the rest of the people: $10!$ ways
- Insert the pair in the arranged line: $11$ ways
- Swap a man and a woman in the pair: $2$ ways
Therefore, the result is $6 \cdot 6 \cdot 10! \cdot 11 \cdot 2 = 11! \cdot 72$
But my friends argue that we don't have to choose a man and a woman in the pair because the problem already told us that there is just $1$ man and $1$ woman that must stand next to each other. My argument is that if we don't choose which pair it is, we can't write the outcomes because we don't know which pair it is.
Which one is the right way?