Question

We are seating 5 people for dinner: A, B, C, D, E

If the chairs are placed in a circle:
How many seating arrangements are possible?
How many if A and B MUST be next to each other?

Answer 1

I will assume that there no designated chairs, so
let's put somebody in a chair. That leaves 4 to be placed around that person
number of ways = 4! or 24

or

look at it this way: the 5 people can be arranged in 5! ways or 120
suppose everybody gets up and moves one chair to the right, we don't
get a new arrangement. This can be done 5 times, so number of ways
= 5!/5 = 4! , same as above

If A and B must sit beside each other, treat them as one unit X
so now we have C, D, E, X

Using the same argument as above, we can seat these in 3! ways
but the X could be AB or BA, with no changes to the rest of the seating
so total number of ways = 2(3!) = 12