Multiset Permutations Calculator

About this calculator

When you arrange n items in a row where some items repeat, the number of distinct arrangements is fewer than n! because swapping two identical items produces the same-looking sequence. The correct count is the multiset permutation formula: n! / (r₁! · r₂! · ... · rₖ!), where r₁, r₂, ... are the counts of each repeated item. The classic textbook example is the word "MISSISSIPPI": with 11 letters where I appears 4 times, S appears 4 times, and P appears 2 times, the number of distinct letter arrangements is 11!/(4!·4!·2!) = 34,650, not 11!.

This formula is a core building block in combinatorics courses and shows up wherever repeated elements need to be arranged: computer scientists use it to count distinct binary strings with a fixed number of 1s and 0s, geneticists use it to count possible sequences of repeated nucleotide patterns, and it's the basis for the multinomial coefficient used in probability distributions over categorical outcomes.

This calculator takes your total item count and the repeat counts for each distinct item and applies the formula directly, so you get the exact number of distinct arrangements without factoring out the repeats by hand.