Binomial Coefficient Calculator (n choose k)
Calculate n choose k exactly, even for huge numbers: C(n, k) with every digit, scientific notation, permutations P(n, k) and the matching row of Pascal’s triangle.
The binomial coefficient C(n, k), read "n choose k", counts how many different groups of k items you can pick from n when order does not matter. It gives lottery odds, poker hand counts, the coefficients in (a + b)ⁿ and the numbers in Pascal’s triangle. This calculator works with whole-number arithmetic, so even results with thousands of digits are exact, not rounded.
How to use this tool
- Enter n, the total number of distinct items.
- Enter k, how many you choose.
- Read C(n, k) with every digit, plus scientific notation for large values.
- Compare with P(n, k) if the order of the chosen items matters.
- For n up to 60, scan the Pascal’s triangle row to see every C(n, k) at once.
Formula
C(n, k) = n! / (k! × (n − k)!)
n! is n factorial (n × (n − 1) × … × 1, with 0! = 1). The calculator multiplies (n − k + 1)/1 × (n − k + 2)/2 × … step by step, which stays a whole number at every step and avoids huge factorials. Permutations are P(n, k) = C(n, k) × k!.
Example
How many ways can a committee of 3 be chosen from 10 people? C(10, 3) = 10! / (3! × 7!) = (10 × 9 × 8) / (3 × 2 × 1) = 120. If the three seats are different roles (chair, secretary, treasurer), order matters and P(10, 3) = 10 × 9 × 8 = 720, which is 120 × 3!.
Frequently asked questions
What are the odds of winning a 6/49 lottery?
There are C(49, 6) = 13,983,816 possible tickets, so a single ticket has a 1 in 13,983,816 chance of matching all six numbers.
What is C(n, 0) or C(n, n)?
Both equal 1: there is exactly one way to choose nothing and one way to choose everything. C(0, 0) is also 1.
What happens when k is larger than n?
You cannot choose more items than exist, so C(n, k) = 0. The calculator shows 0 rather than an error.
How is this different from combinations with repetition?
C(n, k) picks each item at most once. If items can be picked repeatedly, the count is C(n + k − 1, k); use the stars and bars calculator for that.
Last updated: 2026-10-30