When four people sit down at a round table to play cards, two lists of their four names are equivalent as seating charts if each person has the same person to the right in both lists9. (The person to the right of the person in position 4 of the list is the person in position 1). We will use Theorem 1.5 to count the number of possible ways to seat the players. We will take our set S to be the set of all 4-element permutations of the four people, i.e., the set of all lists of the four people.

(a) How many lists are equivalent to a given one?

(b) What are the lists equivalent to ABCD?

(c) Is the relationship of equivalence an equivalence relation?

(d) Use the Quotient Principle to compute the number of equivalence classes, and hence, the number of possible ways to seat the players.

Theorem 1.5:

If an equivalence relation on a p-element set S has q classes each of size r, then q = p/r.