Combinations and Permutations Calculator

You can use this combinations and permutations calculator to quickly and easily calculate the number of potential combinations and permutations of r elements within a set of n objects.

Calculate Combinations and Permutations in Five Easy Steps:

  • 1. Select whether you would like to calculate the number of combinations or the number of permutations using the simple drop-down menu
  • 2. Enter the total number of objects (n) and number of elements taken at a time (r)
  • 3. Select whether repeat elements are permitted
  • 4. Input a list of elements, separated by commas (optional)
  • 5. Press the "Calculate" button to compute the results.
Combinations (nCr) & Permutations (nPr) Calculator

Combinations vs. Permutations

Mathematics and statistics disciplines require us to count. This is particularly important when completing probability problems.

Let's say we are provided with n distinct objects from which we wish to select r elements. This type of activity is required in a mathematics discipline that is known as combinatorics; i.e., the study of counting. Two different methods can be employed to count r objects within n elements: combinations and permutations. These two concepts are very similar and are frequently confused.

The Difference Between a Combination and a Permutation

When considering the differences between combinations and permutations, we are essentially concerned with the concept of order. A permutation relates to the order in which we choose the elements. When the same set of elements are taken in a different order, we will have different permutations. When we are not concerned with order in which we select r elements from a set of n objects, the order is not taken into consideration.

An Example of Permutations

It's worth looking at an example to better differentiate between these two concepts.

First, let's consider how many permutations of two letters there are within the following set of four letters: {A,B,C,D}.

In this case, we list all object pairs from the set while also taking order into consideration. We find that there are 12 permutations in total: AB, AC, AD, BA, BC, BD, CA, CB, CD, DA, DB, and DC.

It is important to recognize that the AB and BA permutations are dissimilar because, in the first case, A was selected first while, in the second, B was selected first; i.e., the order is of significance.

An Example of Combinations

Let's calculate the number of combinations of two letters from the same set: {A,B,C,D}.

As we are calculating combinations, we are no longer interested in the order of the elements. As such, we can quickly and easily identify all the combinations by looking at the permutations and deleting all those that include the same letters.

In this regard, AB and BA are treated as being the same. As such, we have six combinations: AB, AC, AD, BC, BD, and CD.



Permutations and Combinations with / without Repetition
Type Is Repetition Allowed? Formula
r-permutations Yes P(n, r) = nr
r-permutations No P(n, r) = n! / (n - r)!
r-combinations Yes C(n, r) = (r + n - 1)! / (r! * (n - 1)!)
r-combinations No C(n, r) = n! / (r! * (n - r)!)

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