Solution
A Single Gift Chain
Show the problem again
Six friends hold a gift exchange: each person gives exactly one gift to someone else and receives exactly one gift. Call the assignment a single chain if, starting from any person and repeatedly moving to the person they give to, you visit all six people before returning to the start. How many assignments are single chains? Give an integer.
Worked solution
The answer is 120. An assignment is a permutation of the six people, and a single chain is a permutation consisting of one 6-cycle. Starting the cycle at a fixed person, say the first friend, the cycle is determined by the order in which the other five people are visited: 5! = 120 ways. Each 6-cycle is counted exactly once, since its starting point is pinned. More generally there are (n − 1)! cyclic permutations of n objects.
Source: Classic counting exercise in wide circulation; no traceable original source. Statement written for AxiomIQ.