Combination Calculator (nCr)
Count the ways to choose r items from n when order does not matter.
Results update as you type.
About this calculator
A combination counts how many ways you can choose r items from a set of n when order does not matter — picking the same group in a different sequence counts once. This calculator returns the exact count, the formula, and the matching number of ordered arrangements (nPr) for comparison.
The formula is nCr = n! / (r! × (n − r)!). For 10 items taken 3 at a time there are 120 distinct groups. Combinations are always smaller than permutations because each unordered group corresponds to r! different orderings, so nCr = nPr ÷ r!. Choosing none or all gives nC0 = nCn = 1, and as with permutations, r cannot exceed n.
Combinations answer "how many groups?" — the odds in a lottery, the number of possible five-card poker hands, ways to pick a committee, or terms in a binomial expansion. The value nCr is also called the binomial coefficient, the "n choose r" number in Pascal's triangle.
Frequently asked questions
What is the combination formula?
nCr = n! / (r! × (n − r)!). For 10 items taken 3 at a time: 120 different groups.
Why is nCr smaller than nPr?
Combinations ignore order, so each unordered group corresponds to r! ordered permutations. That is why nCr = nPr ÷ r!.
When should I use a combination?
When order does not matter: lottery numbers, a hand of cards, choosing a committee, or any selection where the group is what counts, not its arrangement.
What is a binomial coefficient?
It is another name for nCr, the "n choose r" value that appears in the binomial theorem and forms the entries of Pascal's triangle.
What is nC0 or nCn?
Both equal 1 — there is one way to choose none, and one way to choose all of them.
Does this tool also show permutations?
Yes. It shows nPr = nCr × r! alongside the combination so you can compare ordered and unordered counts.
API — use this calculator from code
Call this calculator as a free JSON endpoint — no key required. Send the field values below as query parameters or JSON. Read the full API docs →
Endpoint
GET https://calculator.free/api/v1/combination/curl
curl "https://calculator.free/api/v1/combination/?n=10&r=3"JavaScript fetch()
const r = await fetch(
"https://calculator.free/api/v1/combination/?" + new URLSearchParams({
"n": "10",
"r": "3"
}));
const data = await r.json();
console.log(data.results);Results are estimates for general guidance only, not financial, medical or tax advice.