Permutation with Repetition Calculator

Created by Krishna Nelaturu
Reviewed by Steven Wooding
Last updated: Aug 25, 2022

Our permutation with repetition calculator will help you determine the different ways you can order your objects. You can also use this calculator to list all permutations possible with your set of objects. In this article, we shall:

  • Define permutation.
  • Learn the formula for permutation with repetition.
  • Look at an example permutation calculation.

What are permutations?

Say you have to set a new 4-digit pin for your device. You can arrange the ten numbers from 0 to 9 into these four places in any order. Each possible arrangement is called a permutation.

Permutation is an arrangement of items in a particular order from a given set of items.

Permutation with repetition: When items in the set are repeated in the arrangement, it is a permutation with repetition. In the above example, pins like 3242 or 4555 have repeated items (2 and 5).

Permutation without repetition: When duplication of items is not allowed in the arrangement, it is a permutation without repetition. Each item in the permutation is unique. In such cases, pins like 3242 are not allowed since 2 occurs twice. However, 4321 is allowed since each digit is unique.

Permutation with repetition formula

The formula for permutation with repetition is:

nPr=nr^nP_r = n^r

where:

  • nn – Number of items in the set;
  • PP – Total number of possible permutations; and
  • rr – Number of items allowed in each permutation.

Read nPr^nP_r as the number of ways to arrange nn items in rr places. Note that other representations such as PrnP^n_r and P(n,r)P(n,r) are also common.

How do you calculate permutation with repetition?

To calculate the number of permutations with repetition when arranging n items in r places, simply multiply n with itself r times. ⁿPᵣ = nʳ.

Going back to our four-digit pin problem, let's work out how many permutations are possible with repetition using the permutation equation:

10P4=104=10, ⁣000^{10}P_4 = 10^4 = 10,\!000

You have 10,000 possible permutations to choose from, so take the opportunity to set a hard-to-guess pin and keep your data safe from prying fingers! Avoid easy-to-guess pins like your birth year or 1234.

How to use this permutation with repetition calculator

Omni strives to make our tools versatile and robust while keeping things user-friendly. You'll find that our permutation with repetition calculator is no exception to this rule:

  • Enter the total number of objects n and the sample size r.
  • The calculator will automatically give you:
    • Permutations with repetition; and
    • Permutations without repetition, so you can compare the two.
  • This calculator will also list all possible permutations if the total number doesn't exceed the maximum display preset of 2000. You can change this value in the advanced mode, but a lengthier list may not be convenient.

Other relevant calculators

We have more calculators related to permutations and combinations for you:

FAQ

How many different 10-character passwords are possible using only alpha numerals?

There are 839,299,365,868,340,224 possible permutations. You can calculate this yourself by approaching this as a permutation with repetition problem:

  1. Determine the total number of items available for permutation:
    • Accounting for upper and lower case English alphabets gives 52 characters.
    • There are ten digits from 0 to 9. Hence we have 62 characters in total.
  2. Calculate the number of ways we can arrange 62 characters into 10 places when repetition is allowed: 62¹⁰ = 839,299,365,868,340,224.
Krishna Nelaturu
Set of objects
Total number of objects n
Sample size r
Number of selections
Permutations with repetitions
Permutations without repetitions
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Permutations with repetitions
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