Permutations with Repetition Calculator
How many arrangements are possible when things can repeat, or when the set you are rearranging already contains duplicates.
Results
What this tool does
The ordinary permutation formula assumes every item is distinct and used once. Two common situations break that. The first is filling slots from a fixed set with repeats allowed — PINs, number plates, passwords — where the answer is simply the number of choices raised to the number of slots. The second is rearranging something that already contains duplicates, where the identical items are interchangeable and the count drops accordingly. This page does both, and shows the plain permutation alongside so the size of the difference is visible.
Formula
with repetition: n^r · rearranging a set with repeats: total! ÷ (product of the factorials of the counts)
Variables
| Symbol | Meaning | Unit |
|---|---|---|
mode | Which kind | — |
n | How many to choose from | — |
r | How many positions to fill | — |
counts | How many of each kind | — |
OUT | Different arrangements | — |
WR | If nothing could repeat | — |
CR | If order did not matter | — |
AD | If every item were distinct | — |
RD | The repeats divide it by | x |
TI | Items in all | — |
DG | Digits it has | — |
Worked example
- Which kindrepeat
- How many to choose from10
- How many positions to fill4
- How many of each kind1, 1, 2, 2, 2, 1, 1, 1
- Different arrangements10,000
- If nothing could repeat5040
- If order did not matter715
- Digits it has5
Frequently asked questions
Which of the two modes do I want?
The first if you are filling positions from a fixed set and may use the same thing more than once: a four-digit PIN is ten choices in each of four positions, so ten to the fourth. The second if you are rearranging a collection that already contains repeats: the letters of MISSISSIPPI are eleven items, but the four S's are interchangeable, so the arrangements come to far fewer than eleven factorial. Write the counts — how many of each distinct kind — and it divides them out.