Combination Calculator

What this calculator does

So, you want to pick a group out of a bigger set, and you do not care what order you pick them in. How many different teams of 3 can you make from 5 people? How many five-card hands are there in a deck? That is a combination, and this counts them for you.

Tell it the total number of items and how many you are choosing, and it gives you the number of possible combinations. It runs right here in the browser.

Using the calculator

1. Enter the total number of items, the size of the set you are choosing from. This is n.
2. Enter the number of items chosen, how many you are picking. This is r.
3. Press Calculate.

It returns the number of possible combinations. Since you cannot choose more items than you have, n needs to be at least as big as r. Reset clears both boxes.

What a combination is

A combination is a selection where order does not matter. If you pick Anna, Ben and Chan for a committee, that is the exact same committee as Chan, Anna and Ben. Shuffling the same people around does not make a new group.

You will see it written a few ways: nCr, or C(n, r), or the two numbers stacked inside a bracket. They all mean the same thing, and they are all read as "n choose r": out of n things, choose r, and count how many different selections you could end up with.

The formula, and what each part means

The number of combinations is:

C(n, r) = n! ÷ ( r! × (n − r)! )

Here n is the total number of items, r is how many you are choosing, and the exclamation mark means factorial, which is just multiplying a whole number down to 1. So 5! = 5 × 4 × 3 × 2 × 1 = 120. The tool works out those three factorials and divides.

Combination vs permutation: the one difference

This is the thing people mix up, and it comes down to a single question: does the order matter?

A gold, silver and bronze podium is a permutation, because first place and third place are not the same thing, so the order counts. A committee of three is a combination, because there are no positions, just a group. Same people, different question.

And the numbers show it. Choosing 3 from 5 people gives 60 possible podiums but only 10 possible committees. The podium count is bigger because it counts every reordering of the same three people as different, while the committee count does not. If order matters for your problem, use the permutation calculator instead.

Worked examples

Choosing 3 desserts from 5. A menu has five desserts and you may pick any three. How many different trios could you order?

C(5, 3) = 5! ÷ (3! × 2!) = 120 ÷ (6 × 2) = 120 ÷ 12 = 10. So there are ten possible trios, few enough that you could list them to check.

The handshake problem. Eight people at a table each shake hands once with everyone else. How many handshakes? A handshake is just a pair of people chosen from the eight, and order does not matter (you shaking my hand is the same handshake as me shaking yours). So it is C(8, 2) = 8! ÷ (2! × 6!) = 40320 ÷ (2 × 720) = 28 handshakes.

Where the formula comes from

It is easier to see if you start with ordered picks and then correct for the ordering. Picking r items from n in order gives n! ÷ (n − r)! arrangements. But you do not want order, and every single group of r items has been counted once for each way of arranging it, which is r! times over. So you divide that overcounting out, and n! ÷ (n − r)! becomes n! ÷ (r! × (n − r)!). That r! in the denominator is the whole difference between a permutation and a combination.

A couple of shortcuts worth knowing

Choosing r is the same as leaving out n minus r. Picking which 3 of 5 people to include is the same job as picking which 2 to leave out, so C(5, 3) equals C(5, 2). Both are 10. In general C(n, r) = C(n, n − r), which is handy: to count "choose 18 from 20", just count "choose 2 from 20" instead, which is far quicker.

The edges are always 1. There is exactly one way to choose none (pick nothing) and exactly one way to choose all of them, so C(n, 0) and C(n, n) are both 1.

A note on large numbers

The tool works the answer out from the factorials directly, which is exact and instant for the values you will normally use. Worth knowing, though, that factorials grow at a frightening pace: 10! is already over three million, and 20! is more than two quintillion. Past roughly n = 18, those numbers get bigger than the range a browser keeps perfectly exact, so for very large n the last digits of the result may not be reliable. For everyday combinatorics it is spot on; for enormous n, treat a huge result as very close rather than exact to the final digit.

Questions people ask

Does the order matter in a combination?

No. That is the defining feature. Picking the same items in a different order gives the same combination. If order matters, you want a permutation.

What is the difference between nCr and nPr?

nCr counts selections where order does not matter; nPr counts arrangements where it does. nPr is always the larger of the two, by a factor of r!.

Why does C(5, 3) equal C(5, 2)?

Because choosing which 3 to keep is the same as choosing which 2 to leave out. In general, C(n, r) = C(n, n − r).

What if I choose more items than there are?

You cannot, so there is no valid combination. The number chosen, r, has to be no bigger than the total, n.

Where are combinations used?

Anywhere you count selections without order: lottery draws, card hands, picking teams or committees, and working out probabilities.

References

A note on where this comes from. The count of ways to choose r items from n without regard to order is a foundational result in combinatorics, the same quantity as the binomial coefficient "n choose r", and it is set out in standard references such as Wolfram MathWorld and the classic text Concrete Mathematics. For further reading, see Combination.

1. Weisstein, Eric W. "Combination", Wolfram MathWorld, on the count of unordered selections and the formula n! / (r!(n − r)!).
2. Graham, Knuth and Patashnik, Concrete Mathematics, on binomial coefficients and the combinatorics of choosing.