Permutação com Repetição
- Created by
- Renato Passos, Eng. de Software
- Reviewed by
- Renato Passos, Eng. de Software
Last updated: Apr 18, 2026
Formula
n!/(Π rᵢ!)
About this calculator
The permutation with repetition calculator computes the number of distinct arrangements of n elements, where some elements are identical. For example, rearranging the letters of the word 'MATHEMATICS' involves repeated letters. The formula used is n! divided by the product of the factorials of the frequencies of each repeated element. This avoids counting identical arrangements as different, ensuring accurate results.
To use the calculator, enter the total number of elements (n) and then the quantities of each type of repeated element. For instance, for the word 'BANANA', n=6, with 3 A's, 2 N's, and 1 B. The calculation would be 6! / (3! * 2! * 1!) = 60 distinct arrangements. This tool is useful in combinatorics problems, anagram analysis, and situations where order matters but there are repetitions.
Use cases include: counting anagrams of words with repeated letters, distributing identical objects into ordered positions, or calculating possibilities in games and passwords with repeated characters. Caution: ensure that the order of elements truly matters; if not, it is a combination. Also, the sum of frequencies must equal n, otherwise the result will be incorrect.
The calculator is based on the permutation with repetition formula, an extension of simple permutation. While simple permutation considers all elements distinct, here we deal with identical elements, reducing the total number of arrangements. It is an essential tool for mathematics students, programmers, and professionals working with combinatorial analysis.
Frequently asked questions
What is the difference between simple permutation and permutation with repetition?
In simple permutation, all elements are distinct, and the number of arrangements is n!. In permutation with repetition, some elements are identical, and the formula is n! divided by the product of the factorials of the frequencies.
How to calculate anagrams of a word with repeated letters?
Use the permutation with repetition formula: divide the factorial of the total number of letters by the product of the factorials of the counts of each repeated letter.
What if the sum of frequencies is not equal to n?
The calculation will be incorrect. The sum of the frequencies of each type of element must exactly equal the total number of elements (n).
Can I use this calculator for passwords with repeated characters?
Yes, as long as order matters. For example, 4-digit passwords with two identical digits and two different ones can be calculated using this formula.
Does the calculator work for large numbers?
Yes, but factorials of very large numbers can produce huge results. The calculator handles big integers, but there may be precision limitations in some cases.