In this previous question, it was asked how many different ways we can arrange 4 managers and 3 employees in 7 seats around a circular table. One user said that there were 144 ways. I said there were 16. Whose answer is correct? Are we both wrong? Is it dependent on what is meant by "arrange"?
It depends on what you mean by arrange. Do you think all managers are interchangeable and all employees are interchangeable, or is an arrangement where you swap the seats of two employees different? Are the seats numbered, so if we rotate everybody one place we get a different arrangement? How about if we mirror the arrangement, so clockwise becomes counterclockwise-is that different?
If the people are interchangeable and the seats are not numbered, we must have a subsequence $EMME$ because two managers must sit together (there aren't enough employees to separate them) and a third is not permitted. The only arrangements are then $MMEMMEE, MMEMEME$-$2$ of them. If the seats are numbered, each of these can be rotated seven ways, giving $14$ arrangements. If the seats are not numbered but the people are distinguishable, we can arrange the managers $4!=24$ ways and the employees $3!=6$ ways for each order, giving $2\cdot 24 \cdot 6=288$ ways. Finally if the people are different and the seats are numbered, we multiply by $7$ to get $2016$ arrangements.
For the record, the correct answer is $288$ (nobody had found it before I just answered the original question; Ross Millikan was close but his arithmetic fell prey to the gravitational attraction of a previous wrong answer).