Binomial Coefficient Calculator

The binomial coefficient C(n, k), also written as "n choose k" or as the symbol ⁿCₖ, is the number of ways to select exactly k items from a set of n distinct items when the order of selection does not matter. It is one of the most fundamental quantities in combinatorics and appears throughout probability theory, algebra, statistics, and computer science. The formula for the binomial coefficient is C(n, k) = n! / (k! × (n − k)!), where the exclamation mark denotes the factorial function: n! = n × (n−1) × (n−2) × ⋯ × 2 × 1, and 0! = 1 by convention. For example, C(5, 2) = 5! / (2! × 3!) = 120 / (2 × 6) = 10, meaning there are exactly 10 ways to pick 2 items from a group of 5. Binomial coefficients are the entries of Pascal's triangle. In Pascal's triangle, each number is the sum of the two directly above it. The entry in row n and column k (counting from zero) is exactly C(n, k). This property follows from Pascal's identity: C(n, k) = C(n−1, k−1) + C(n−1, k), which says that each item in the selection is either included (reducing the problem to choosing k−1 from n−1) or excluded (reducing it to choosing k from n−1). The name "binomial coefficient" comes from the binomial theorem: (x + y)ⁿ = Σ C(n, k) × xᵏ × y^(n−k) for k from 0 to n. The coefficient of each term xᵏ y^(n−k) in the expansion of (x + y)ⁿ is exactly C(n, k). For example, (x + y)³ = x³ + 3x²y + 3xy² + y³, and the coefficients 1, 3, 3, 1 are C(3,0), C(3,1), C(3,2), C(3,3). In probability, binomial coefficients arise in the binomial distribution, which models the number of successes in n independent Bernoulli trials each with success probability p. The probability of exactly k successes is C(n, k) × p^k × (1−p)^(n−k). Combinatorics problems such as counting poker hands, lottery tickets, committee selections, or binary strings with a fixed number of ones all reduce directly to binomial coefficient calculations. For large n and k, direct factorial computation causes integer overflow. Efficient algorithms compute C(n, k) iteratively using the multiplicative formula C(n, k) = ∏ (n − i) / (i + 1) for i from 0 to k−1, which keeps intermediate values smaller. This calculator uses exact integer arithmetic to return precise results for all practical inputs.