Combinations with Repetition Calculator

Created by Luis Hoyos
Reviewed by Wojciech Sas, PhD candidate
Last updated: Apr 11, 2022

In this article, we deal with how to calculate combinations with repetition.

Combinations are present in many everyday life situations, such as when wanting to know the chances to win the main prize in a lottery (check more about it in our lottery calculator). For that reason, we've created this and other similar calculators.

The easiest way to calculate combinations with repetition is using our calculator, but knowing the equation behind the results is helpful for a deeper understanding. For that reason, in the next section, we present the formula for combinations with repetition and an example of calculation.

🙋 This tool doesn't only calculate the number of combinations with repetitions but also serves as a combination generator with repetition, showing you the list of all those possible combinations.

Don't confuse combinations with permutations.

Combination with repetition formula

This formula gives the number of combinations with repetition:

C(n,r)=(r+n1)!r!(n1)!,\footnotesize C'(n,r) = \frac{(r+n-1)!}{r!(n-1)!},

where:

  • rr — The sample size;
  • nn — The total number of objects; and
  • !! — The factorial of a number.

Example of combination with repetition

Suppose you want to know how many combinations of 5 numbers with repetition are possible. Follow these steps to get the answer:

  1. Identify the total number of objects (n) and the sample size (r). There exist ten digits in the numeric system (from 0 to 9), and we want five-numbers combinations. Therefore, n=10n = 10 and r=5r = 5.
  2. Input the values in the combination with repetition formula:
    C(n,r)=(r+n1)!r!(n1)!)=(5+101)!5!(101)!=14!5!×9!=87,178,291,200120×362,880=2002C'(n,r) = \frac{(r+n-1)!}{r!(n-1)!}) = \frac{(5+10-1)!}{5!(10-1)!} = \frac{14!}{5!×9!} = \frac{87,178,291,200}{120×362,880} = 2002
  3. That's it. The number of combinations with repetition with five numbers is 2002. You can check these results in our combinations with repetition calculator.

Other Omni tools similar to the combinations with repetition calculator

You can generate combinations with repetition and many other measures using these calculators:

FAQ

How many combinations with 5 numbers with repetition are possible?

There are 2,002 possible combinations with repetitions and 100,000 permutations with repetitions of arranging the digits 0-9 to form a five-digit number.

How to calculate combinations with repetition?

To calculate combinations with repetition C'(n,r):

  1. Identify the total number of objects (n)
  2. Establish or identify the sample size (r)
  3. Input the values in the following formula:
    C'(n,r) = (r+n-1)! / (r!(n-1)!).
Luis Hoyos
Set of objects
Total number of objects n
Sample size r
Number of selections
Combinations with repetitions
Combination generator
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Combinations with repetitions
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