Binomial Coefficient Calculator
Compute the binomial coefficient 'n choose k' — the number of ways to pick k items from a set of n when order does not matter. Enter n and k and this calculator returns C(n, k) = n! / (k!·(n−k)!).
Formula
C(n, k) = n! ÷ (k! × (n − k)!)
- C(n, k), read 'n choose k', counts the number of ways to select k items from n distinct items when order does not matter.
- n and k are rounded down to whole numbers, since you cannot choose a fractional number of items.
- If k is negative, greater than n, or n is negative, the result is 0 — there are no valid selections.
- C(n, k) always equals C(n, n−k): choosing which k to include is the same as choosing which n−k to leave out.
- These coefficients are exactly the numbers in row n of Pascal's triangle.
10 choose 3
Inputs
- n: 10
- k: 3
C(10, 3) = 10! / (3!·7!) = (10×9×8) / (3×2×1) = 720 / 6 = 120. There are 120 ways to choose 3 items from 10.
Frequently asked questions
What does 'n choose k' mean?
It is the number of different ways to pick a group of k items from a larger set of n items, where the order you pick them in does not matter.
How is it different from a permutation?
Permutations count ordered arrangements, so they are larger. The binomial coefficient counts unordered selections — it divides out the k! ways each group could be ordered.
Why is C(n, k) equal to C(n, n−k)?
Choosing the k items to include automatically determines the n−k items left out, so the two counts are always identical.
What if k is bigger than n?
You cannot choose more items than exist, so the result is 0. The same applies to negative inputs.
How does this relate to Pascal's triangle?
Each entry in Pascal's triangle is a binomial coefficient: row n, position k equals C(n, k).