Combinatorics Visual Tools

11Tools

2Categories

100%Free

5 tools

Permutations

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.

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)!

Partial Permutation Visualizer

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.

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.

Permutation with Repetition Visualizer

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.

6 tools

Combinations

Simple Combination Visualizer

Pick r items from n where order doesn

Distribution into Cells Visualizer

Send n distinct items into k labeled cells where each item picks its own cell — watch the assignment tuple fill in below the cells and the k^n outcomes group by item 1

Partition into Groups Visualizer

Split n distinct items into k labeled boxes of fixed sizes — watch each partition build position by position and group by which box receives item 1. Cycle the partition shape to see how the multinomial formula n!/(n1!·n2!·...) reshapes the count.

Pascal

Click any cell to focus a binomial coefficient C(n, r) and switch between four identity modes — Pascal

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.