Combinatorics Calculator
Work out the number of combinations or permutations when choosing k out of n elements.
How to use the combinatorics calculator
- Enter n (total number of elements) and k (number to be chosen).
- Choose whether order matters — combinations (no) or permutations (yes).
- The calculator shows the count automatically as you type.
How combinatorics is calculated
For combinations, where order doesn't matter, the notation is:
Worked example: the number of ways to choose 3 from 10 when order does not matter is 10! ÷ (3! × 7!) = 120. If order does matter, we are dealing with permutations instead: 10 × 9 × 8 = 720, six times as many. The factor of six comes from the same three elements being arrangeable in 3! = 6 ways. The question «does order matter?» is therefore the only thing you need to settle first.
Example: "How many ways can you choose 3 out of 10 people for a team?" — here it doesn't matter what order the 3 are chosen in, so you use combinations.
For permutations, where order matters, you instead use:
Example: "How many ways can 3 out of 10 people finish 1st, 2nd and 3rd?" — here the order matters completely, so you use permutations.
Frequently asked questions
What's the difference between combinations and permutations?
With combinations, order doesn't matter (choosing A, B, C is the same as B, C, A), while with permutations, order counts as different outcomes. Permutations therefore always give an equal or greater count than combinations for the same n and k.
Why do the numbers get so large?
Because the number of possibilities grows very quickly with n and k — this is called combinatorial explosion. The calculator uses a method that avoids computing factorials directly, so it can handle large numbers without breaking.
How is this used in practice?
Combinatorics is used, among other things, to work out probabilities in lotteries and card games, the number of possible passwords, or the number of ways to organize a group of people.