Permutation with Repetition Calculator
Count ordered arrangements when items can repeat (n^r, like PINs) or when a word has repeated letters (MISSISSIPPI), with exact results.
“Permutation with repetition” covers two everyday counting problems. When items can be reused, such as digits in a PIN, letters in a code or repeated dice rolls, the count is n^r. When you rearrange a fixed set that already contains duplicates, such as the letters of MISSISSIPPI, the count is N! divided by the factorial of each repeat. This calculator handles both, with exact results however large they get.
How to use this tool
- Pick the problem type: items can be reused, or arranging a word or set with duplicates.
- For n^r, enter how many choices each position has (n) and how many positions there are (r).
- For duplicates, type the word, or the count of each repeated item, such as 4, 4, 2, 1.
- Read the exact count and compare it with the no-repetition figure.
Formula
Items reused: n^r · Arranging duplicates: N! / (n₁! × n₂! × … × nₖ!)
In n^r each of the r positions can be any of the n options, so the choices multiply. In the multiset formula, N is the total number of items and n₁ … nₖ are how many times each distinct item appears; dividing by each nᵢ! removes the swaps of identical items, which do not create a new arrangement.
Example
A 4-digit PIN uses the digits 0–9 with repeats allowed: 10^4 = 10,000 codes, against 10 × 9 × 8 × 7 = 5,040 if no digit could repeat. The word MISSISSIPPI has 11 letters: M once, I four times, S four times and P twice, so 11! / (1! × 4! × 4! × 2!) = 39,916,800 / 1,152 = 34,650 distinct arrangements.
Frequently asked questions
How many 3-letter codes can be made from A to Z?
26^3 = 17,576 if letters can repeat, and 26 × 25 × 24 = 15,600 if they cannot.
Is this the same as combinations with repetition?
No. Permutations count order, so AB and BA are different. Combinations with repetition ignore order and use C(n + r − 1, r), which is much smaller.
How many ways can the letters of BANANA be arranged?
6! / (3! × 2! × 1!) = 720 / 12 = 60, because A appears three times and N twice.
Are capital and lower-case letters treated differently?
No. In word mode letters are counted ignoring case and spaces, so “Mississippi” and “MISSISSIPPI” give the same answer. Type the counts instead if you need a different grouping.
Last updated: 2026-10-30