Three counts for three questions
This tool reports three distinct combinatorial counts. The factorial n! is the number of ways to arrange all n distinct items in a row. Permutations with repetition, nʳ, count ordered sequences of length r when each of the n options can be reused freely — the classic example is a PIN, where each position independently has n choices. Combinations with repetition count unordered selections that allow repeats, such as picking r scoops from n flavours where you may take the same flavour twice.
The chart places the three side by side so you can see how allowing repetition and caring about order each push the count higher.
Where the formulas come from
Permutations with repetition follow the multiplication principle: with r independent positions and n choices each, the total is n × n × … × n, or nʳ. Combinations with repetition use the "stars and bars" argument, which gives C(n + r − 1, r) — the number of ways to distribute r identical picks across n categories.
- n! — arrangements of all n distinct items, no repetition.
- nʳ — ordered, repetition allowed (codes, PINs, license plates).
- C(n + r − 1, r) — unordered, repetition allowed (multisets, scoops, dice totals).
Choosing the right model
The two questions to settle first are whether order matters and whether items can repeat. Order plus repetition points to nʳ; no order plus repetition points to the combinations-with-repetition count; if items cannot repeat at all, switch to the standard permutation and combination calculator.
A practical caution: because n! and nʳ grow explosively, large inputs can exceed the precision of exact whole-number arithmetic, after which the displayed counts become approximate.
Formula
n!; permutations with repetition = nʳ; combinations with repetition = C(n+r−1, r)Frequently asked questions
- When do I allow repetition?
- Allow repetition when the same item can be picked more than once, such as digits in a PIN or scoops in an ice-cream cone. Use the standard permutation/combination calculator when each item is used at most once.