Circular Permutation Visualizer

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.

Simple Combination Visualizer

Pick r items from n where order doesn

Diagrams

Key Terms

Full Permutation Visualizer

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)!

Key Terms

Key Terms

Key Terms

Permutation with Identical Items Visualizer

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.

Key Terms

Strong Composition Visualizer

Drop k−1 bars into the n−1 gaps between identical items to split them into k nonempty bins — every bin must hold at least one. The gap-selection encoding makes C(n−1, k−1) visible directly, with live bin brackets and a chosen-gaps readout updating as the bars land.

Weak Composition Visualizer

Drop k−1 bars into a strip of n+k−1 cells to split n identical items into k bins — empty bins allowed. The stars-and-bars encoding makes C(n+k−1, k−1) visible directly, with live bin brackets, a composition tuple readout, and bar-position readout updating as the bars land.

Permutations Visualizer