Great question! I'm happy to help!
You are SO close --- in fact, if we were seating these five knights in a straight line (for example, on a long bench), then the number of arrangements would be 5! = 120
Here's what's diabolically tricky about this question: they're at a ROUND table.
Suppose the 5 knights are A, B, C, D, and E.
If they sit as ABCDE vs.CDEAB on a bench, those are two very different arrangements. At a round table though (assuming there are five seats equally spaced around the table), those two are identical arrangements: each knight has the same two neighbors on his same respective sides. These two seating plans are shown in the diagram below.
So, really the question is: let's say A sits in a seat --- that could be any seat, since they are all symmetrical. Once A has sat, there are 4! ways for the other four knights to fill those other four spaces. Thus, only 4! possibilities.
I hope that was clear and helpful. Please let me know if you have any questions.
knights' seating plans.PNG [ 25.5 KiB | Viewed 489 times ]
Magoosh Test Prep