Suppose King Arthur and the knights of the round table are sitting at the round

Question

Suppose King Arthur and the knights of the round table are sitting at the round table. Assume that there are n chairs and n knights including King Arthur. (a) If one of the chairs is a throne and King Arthur sits in the throne, how many seating arrangements are there? (b) If anyone could sit in the throne, how many seating arrangements are there? (c) If all n chairs are identical, how many seating arrangements are there? (Assume that if everyone moves one seat clockwise, then the seating arrangement has not changed.) (d) In part (c), what would be a good way to uniquely describe a seating arrangement?

Explanation / Answer

(a) Since his majesty's seat is fixed, we are left with n-1 seatings and n-1 people.

The first person has n-1 choices..

The second person has n-2 choices and so on

=> Total combinations = (n-1)*(n-2)*(n-3)*.....1 = (n-1)!

(b) Here there are n chairs including the throne and n people.

So total combinations = n!

(c) First let us fix a person's seat. Then as in problem (a) the number of combinations = (n-1)!

Now if this person would take a different seat, then there won't be any difference since the chairs are identical.

So the number of combinations = (n-1)!

(d) The unique way to describe a seating arrangement would be with respect to King Arthur.

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