Problem:
Four ambassadors and one advisor for each of them are to be seated at a round table chairs numbered in order from to . Each ambassador must sit in an even-numbered chair. Each advisor must sit in a chair adjacent to his or her ambassador. There are ways for the people to be seated at the table under these conditions. Find the remainder when is divided by .
Solution:
We begin by computing the total number of ways to seat the four ambassador–advisor pairs. Each ambassador must sit in one of the six even-numbered seats, with their advisor directly clockwise.
There are ways to choose and order four distinct even-numbered seats for the ambassadors. For each pair, the ambassador and advisor can be labeled in 2 ways, giving total labelings. Therefore, the total number of configurations is
.
We now subtract the number of invalid configurations, which occur when two advisors are assigned to the same seat (i.e., when two ambassador pairs sit in adjacent even-numbered seats).
Choose 2 out of the 4 pairs to sit adjacently in such a way that their advisors conflict. There are such choices. There are 6 adjacent even seat pairs around the circle, and 2 ways to assign the chosen pair to these seats, giving ways. The remaining 2 pairs are placed in the remaining even seats in ways, and their advisors can be assigned in 4 ways. So the number of conflicting arrangements is
.
However, we have overcounted the scenario where all four pairs cause advisor conflicts. In this case, there are distinct ambassador placements (from symmetry and adjacency), and ways to assign ambassadors to those seats, giving configurations that were subtracted too many times.
Thus, the correct total is , and taking we get .
The problems on this page are the property of the MAA's American Mathematics Competitions