Circular Permutation

When objects are arranged in a circle, the usual idea of a "first" position vanishes — rotating the entire circle gives the same arrangement. This is the core insight behind circular permutations: we fix one object to break the symmetry, then arrange the rest. For $n$ distinct objects around a circle, the number of distinct arrangements is $(n-1)!$, not $n!$. This simple shift in perspective often surprises students, but once you see …