KA-0166Counting up to symmetry
5 points, difficulty 3 of 3Level 9–10about 150sSix friends sit around a circular table. Two seatings count as the same if one can be turned into the other by rotating the table. How many genuinely different seatings are there?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
Rotating a circular seating does not change who is next to whom.
2The strategy
There are 6! = 720 ways to seat six people in labelled chairs. Each circular seating appears once for every rotation of the table.
3The full solution
There are 6 rotations, so the answer is 720 / 6 = 120.
Solution
The reliable way
Start by pretending the chairs are labelled: that gives 6! = 720 arrangements. But turning the table produces a seating that the question calls the same, and there are 6 rotations of a six-seat table, so every genuinely different seating has been counted 6 times. Dividing gives 720 / 6 = 120. An equivalent and often faster route: seat one person first to kill the rotation, then arrange the remaining five in 5! = 120 ways. The transferable idea: to count up to a symmetry, either divide by the number of symmetries or fix one object to remove them.
The elegant way
Fix one friend's seat to remove the rotations, then the other five can be arranged in 5! = 120 ways.
Why this is on the test: Counting up to symmetry is exactly where a correct count becomes a wrong answer, and fixing one object is the cleanest fix.