This combination calculator is a tool that will help you determine the number of combinations in a set (often denoted as nCr). Read on to learn:

  • the combination formula,
  • how to calculate combinations and
  • the relation between combination and permutation.

Combination definition

Combination is the number of ways, in which you can choose r elements out of a set containing n distinct objects (that's why such problems are often called "n choose r" problems). The order of choosing the elements is not important.

For example, imagine a bag filled with twelve balls, each one of a different color. You pick five balls at random. How many distinct sets of balls can you get?

Combination formula

Luckily, you don't have to write down all of the possible sets. How to calculate the combinations, then? You can use the formula that will allow you to determine the number of combinations in no time:

C(n,r) = n!/(r! * (n-r)!)

  • C is the number of combinations,
  • n is the total number of elements in the set,
  • r is the number of elements you choose from this set.

The expression on the right hand side is also known as the binomial coefficient.

The exclamation mark represents a factorial. Check out our factorial calculator for more information on this topic.

You can notice that according to the combinations formula, the number of combinations for choosing one element is simply n. On the other hand, if you have to choose all elements, there is only one way to do it.

Let's apply this equation to our problem with colorful balls. We need to determine how many different combinations are there:

C(12,5) = 12!/(5! * (12-5!)) = 12!/(5! * 7!) = 792

You can check the result with our nCr calculator. It will list all possible combinations, too!

Combination and permutation

If you switch on the advanced mode, you will be able to find the number of permutations as well. Permutations are also ways to choose r out of n elements. Unlike in combinations, the order in permutations does matter.

For example, imagine that you have a deck of nine cards with digits from 1 to 9. You draw three random cards and line them up on the table, creating a three-digit number. How many distinct numbers can you create?

If you know the number of combinations, you can easily calculate the number of permutations, too:

P(n,r) = C(n,r) * r!

What next?

Make sure to try our probability calculator if you are interested in probability problems.

Bogna Haponiuk

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