Binomial Coefficient Calculator — C(n, k) Combinations
Enter n (total items) and k (items chosen) to calculate C(n, k) = n! / (k! × (n−k)!) — the number of unordered combinations — plus the row of Pascal's triangle for that n.
Number of ways to choose k items from n without regard to order
- 1
Factor 1: × (10 − 1 + 1) ÷ 1
1 × 10 ÷ 1 = 10 - 2
Factor 2: × (10 − 2 + 1) ÷ 2
10 × 9 ÷ 2 = 45 - 3
Factor 3: × (10 − 3 + 1) ÷ 3
45 × 8 ÷ 3 = 120
How does this calculator work?
C(n, k) = n! ÷ (k! × (n − k)!) counts unordered k-element subsets of an n-element set. Equivalently, multiply (n × (n−1) × … × (n−k+1)) ÷ (k × (k−1) × … × 1). C(n, k) = C(n, n−k) by symmetry, and the sum of all C(n, i) for i = 0 to n equals 2ⁿ.
Formula
How this is calculated
The binomial coefficient C(n, k), read "n choose k", counts the number of ways to select k items from a set of n without caring about order. For example, C(5, 2) = 10 because there are 10 ways to pick 2 items from {A, B, C, D, E}. The formula is n! ÷ (k! × (n − k)!), but computing large factorials directly is slow and causes overflow. This calculator uses the equivalent multiplicative form — multiplying and dividing one pair of factors at a time — which stays numerically stable for moderate n values.
Two important symmetry properties: C(n, k) = C(n, n − k), so choosing 3 from 10 gives the same count as excluding 3 from 10 (leaving 7). Also, the sum of all C(n, k) for k = 0 to n equals 2ⁿ — the total number of subsets of any size. These values form row n of Pascal's triangle, where each entry is the sum of the two entries directly above it in the previous row.
For very large n (above roughly 1000) the result may overflow JavaScript's floating-point range and the calculator returns no output. For exact integer arithmetic with arbitrarily large n, a BigInt-based factorial calculator is needed.
Frequently asked questions
Combinations C(n, k) count selections where order does not matter (choosing a committee of 3 from 10 people). Permutations P(n, k) count arrangements where order matters (choosing a president, vice-president and treasurer from 10 people). P(n, k) = C(n, k) × k!, so permutations are always ≥ combinations for the same n and k.
Pascal's triangle is an infinite triangular array where each row n lists all the binomial coefficients C(n, 0), C(n, 1), …, C(n, n). Every interior entry equals the sum of the two entries directly above it. Row 0 is just [1]; row 4 is [1, 4, 6, 4, 1]. The triangle encodes many combinatorial identities and appears in probability theory, algebra and number theory.
C(n, k) is the coefficient of xᵏ in the expansion of (1 + x)ⁿ (the binomial theorem, hence the name). It appears in probability (binomial distribution), combinatorics (counting subsets), statistics (confidence intervals via the normal approximation to the binomial) and algorithms (dynamic programming, lattice path counting).
TG we-Calculate Editorial Team. (2026). Binomial Coefficient Calculator — C(n, k) Combinations [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/binomial-coefficient-calculator
TG we-Calculate Editorial Team. "Binomial Coefficient Calculator — C(n, k) Combinations." TG we-Calculate. 2026. https://we-calculate.com/calculator/binomial-coefficient-calculator.
TG we-Calculate Editorial Team, "Binomial Coefficient Calculator — C(n, k) Combinations," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/binomial-coefficient-calculator
@misc{wecalculate_binomial_coefficient_calculator, title = {Binomial Coefficient Calculator — C(n, k) Combinations}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/binomial-coefficient-calculator}}, year = {2026}, note = {TG we-Calculate} }
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