Mike and fred are at a conference and have been broken into a group with 10 others people. There is 2 tables that are circular, where one seats 7 and the orher has exactly enough to set the rest. How many different ways can the group be seated if mike and fred must sit together??
consider Mike and Fred as one person.
At the table for 7, then there are
2*11P6 ways to seat them, since the 7th seat is always taken.
and 5! ways to seat the others.
If they do not sit at the big table, then there are
7! ways to seat that table and
2*4! ways to seat the other five.
why the 2*? because Mike and Fred could switch places with each other.
Now, if rotations around the table count as the same order, then the whole ball game changes.