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Combination Calculator: How Many Subsets Are Possible?

Written by:
PK

Below is a combination calculator, which will calculate the number of combinations, or sets you can choose from a larger whole. Enter the total things in the set n and the number you need in your sample r and we'll compute the number of combinations.

If you care about the order of the selection, use the permutation calculator (or change the input in the tool).

Combination Calculator

What is a Combination?

A combination is a unique subset chosen from a larger whole. A combination doesn't care about the order of the elements you choose. That is, if you chose A and B from a set of A, B, and C, it's the same as picking B and A.

Combinations vs. Permutations

Combinations are related to permutations. In permutations you do care about the order of a set, however.

As I joked in the permutation calculator, "a combination lock is a lie". If your combo is 4-26-3 you have to enter it in exactly that order to open the lock. They should be called permutation locks.

Formula for Combinations

The formula for the calculating combinations is:

\frac{n!}{(r!)*(n-r)!}

Where:

  • n – the size of the set
  • r – the size of the subset you are choosing

Compared to the permutation formula, there is an extra term for the factorial of the subset in the denominator.

Example Combination Calculation

Let's say you have 4 letters: A, B, C, D

How many combinations are there when choosing a 2 letter subset?

\frac{4!}{(2!)*(4-2)!}=\frac{24}{2!*2!}=\frac{24}{4}=6

This is easy to verify. The only possible 2 letter subsets from A, B, C, and D are:

  • AB AC AD
  • BC BD
  • CD

There's no other way to choose combination subsets. For example, DC is the same as CD.

Using the Combination Calculator

To compute the total number of combination, first enter "n", the total number of things in your set. Next, enter "r" which is how large of a subset you would like to calculate.

After that, leave the "Permutations or Combinations?" pulldown menu on Combinations, unless you do care about order.

Hit Calculate and we'll let you know how many combinations are possible.

Having fun? Visit our other calculators and tools.

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