Binomial Coefficient Calculator (n choose k)
Compute “n choose k” — the binomial coefficient C(n, k) — the number of ways to pick k items from n where order does not matter.
How it works
C(n, k) = n! / [ k! · (n − k)! ]
The binomial coefficient counts unordered selections — committees, lottery picks, poker hands — where choosing A then B is the same as choosing B then A. It equals the ordered count P(n, k) divided by k! to cancel the orderings of the chosen group. The symmetry C(n, k) = C(n, n−k) reflects that choosing which k to include is the same as choosing which n−k to leave out.
Worked example
C(10, 3) = 10! / (3! · 7!) = (10 × 9 × 8) / (3 × 2 × 1) = 720 / 6 = 120 ways.
Sources: Binomial coefficient — Wikipedia
Frequently asked questions
What is the difference between a combination and a permutation?
A combination ignores order (C(n, k)); a permutation counts order (P(n, k) = C(n, k) × k!). Choosing 3 of 10 gives 120 combinations but 720 ordered permutations.
Why does C(n, k) equal C(n, n−k)?
Every way of picking the k items to keep corresponds to exactly one way of picking the n−k items to leave out, so the two counts are always equal.
What are C(n, 0) and C(n, n)?
Both equal 1: there is exactly one way to choose nothing and one way to choose everything.
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This calculator is for educational purposes. Double-check important results before acting on them.