Watch (n-1)! circular permutations build one by one — one item anchored at the top breaks the rotational symmetry while the remaining items cycle through every distinct seating around the circle. Adjust n and step through the build animation.
Build every n! permutation of n distinct items one ball at a time. Each arrangement starts with a different first item, and rows group all permutations sharing that starter — so you literally see why n! splits as n × (n−1)!
Pick r items from n and arrange them in order, one ball at a time. Each partial permutation is grouped by its first item — making the n × (n−1) × … × (n−r+1) structure of P(n,r) visible at a glance.
Cycle through six multisets — AAB, AAAB, AABB, AABC, AAABB, AABBC — and watch every distinct permutation build one position at a time. Only the specific copy in use dims in the source row, exposing why n! divides by k! for each repeated group.
Build every n^r arrangement of r positions where each spot can be any of n items — source balls never dim because copies are unlimited. Cycle n and r to compare n^r against n!/(n−r)! and see how reuse multiplies the count.