Combination Calculator (nCr)

Calculate the number of ways to choose r items from a set of n, where order doesn't matter.

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n (total items)

r (items chosen)

Result

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Summary

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Detail

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Calculate the number of ways to choose r items from a set of n, where order doesn't matter.

How It Works

How Combination Calculator (nCr) Works

A combination counts how many distinct groups you can form by picking r items out of n total, without caring about the order you picked them in — computed as n! divided by r! times (n−r)!.

Worked Example

See It In Action

Choosing 2 people from a group of 5 (C(5,2)) gives 10 possible pairs, since the order you pick the two people in doesn't create a new pair.

Real-World Use Cases

Who Uses Combination Calculator (nCr) and Why

Working out how many distinct lottery number combinations are possible for a given set of numbers drawn.
Calculating how many different teams or committees of a certain size can be formed from a larger group.
Figuring out how many distinct card hands are possible when dealt a fixed number of cards from a deck.

Common Mistakes

Mistakes to Avoid

Using this calculator when the order of selection actually matters (like assigning distinct roles or ranking outcomes) — that scenario calls for the Permutation Calculator instead, since combinations treat differently-ordered picks as the same result.
Entering r greater than n — you can't choose more items than exist in the set, and the calculation isn't meaningful in that case.
Forgetting that C(n,r) counts groups, not sequences — this matters when deciding whether your real-world problem is really asking for combinations or permutations.

Pro Tips

Tips for Best Results

C(n,r) always equals C(n, n−r) — choosing which r items to include is equivalent to choosing which n−r items to leave out, a useful symmetry for a quick sanity check.
If your scenario involves any notion of "first," "second," or ranked positions, you likely want the Permutation Calculator instead of this one.

Troubleshooting

Fixing Common Problems

I'm not sure whether my problem needs combinations or permutations. — Ask whether swapping the order of two selected items would count as a different outcome. If yes, use the Permutation Calculator; if the group is considered the same regardless of order, use this Combination Calculator.

Glossary

Terms Explained

Combination: A selection of r items from a set of n where the order of selection doesn't matter.

Factorial (n!): The product of all positive integers up to n, used in both the combination and permutation formulas.

FAQ

Frequently Asked Questions

What's the difference between this and a permutation?

Combinations ignore order (picking Alice then Bob is the same group as Bob then Alice), while permutations treat different orderings as distinct outcomes — use the permutation calculator when the sequence itself matters.