Enter values for n (total items) and k (items to choose) to calculate the binomial coefficient — also written as C(n, k) or "n choose k". The calculator returns the binomial coefficient value, along with the factorial breakdown showing exactly how the result is computed. Useful for combinatorics, probability, and statistics problems.
C(n, k) — Binomial Coefficient
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n! (n Factorial)
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k! (k Factorial)
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(n − k)! Factorial
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log₁₀ C(n, k)
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A binomial coefficient C(n, k) — read as 'n choose k' — represents the number of ways to select k items from a set of n distinct items without regard to order. It appears in the binomial theorem and throughout combinatorics and probability theory.
The formula is C(n, k) = n! / (k! × (n − k)!), where '!' denotes factorial. For example, C(10, 3) = 10! / (3! × 7!) = 120. This calculator shows the full factorial breakdown so you can verify each step.
'n choose k' is simply another way of saying the binomial coefficient C(n, k). It answers the question: in how many distinct ways can you choose k items from a collection of n items, ignoring the order of selection?
When k > n, it is impossible to choose k items from a set of only n, so C(n, k) = 0 by definition. The calculator will return 0 in this case.
Both equal 1. C(n, 0) = 1 because there is exactly one way to choose nothing, and C(n, n) = 1 because there is exactly one way to choose all items in the set.
No. A permutation P(n, k) counts ordered selections, while C(n, k) counts unordered selections (combinations). The relationship is C(n, k) = P(n, k) / k!, meaning combinations divide out the k! ways the selected items could be arranged.
Pascal's Triangle is built entirely from binomial coefficients. The nth row (starting from row 0) contains the values C(n, 0), C(n, 1), …, C(n, n). Each entry is the sum of the two entries directly above it, which corresponds to the identity C(n, k) = C(n−1, k−1) + C(n−1, k).
For moderate values of n (up to a few thousand), the calculator uses a logarithm-based approach to avoid overflow, returning the log₁₀ of the coefficient alongside the exact integer result. For very large n, JavaScript's BigInt arithmetic is used to preserve exactness.