Calculate binomial coefficients C(n,k), Pascal's triangle rows, permutations P(n,k), and multinomial coefficients with step-by-step factorial breakdowns.
The binomial coefficient, written as C(n, k) or "n choose k," counts the number of ways to select k items from a set of n distinct items without regard to order. It appears throughout mathematics — in the Binomial Theorem for expanding (a + b)^n, in probability when calculating the chance of k successes in n trials, and in Pascal's triangle where each entry is a binomial coefficient.
The formula is C(n, k) = n! / (k! · (n − k)!). For example, C(5, 2) = 120 / (2 · 6) = 10, meaning there are exactly 10 ways to pick 2 items from 5. This calculator computes not only C(n, k) but also the related permutation count P(n, k) = n! / (n − k)!, the full Pascal's triangle row for n, and multinomial coefficients when you need to split items into more than two groups.
Understanding binomial coefficients is essential for students of algebra, probability, statistics, and computer science. Whether you're expanding polynomial expressions, counting lottery combinations, or analyzing algorithm complexity, this tool gives you instant answers with detailed breakdowns of every factorial involved.
Binomial Coefficient Calculator helps you solve binomial coefficient problems quickly while keeping each step transparent. Instead of redoing long algebra by hand, you can enter n (total items), k (items chosen), k₂ (optional, for multinomial) once and immediately inspect n!, k!, (n−k)! to validate your work.
This is useful for homework checks, classroom examples, and practical what-if analysis. You keep the conceptual understanding while reducing arithmetic mistakes in multi-step calculations.
C(n, k) = n! / (k! × (n − k)!); P(n, k) = n! / (n − k)!; Multinomial = n! / (k₁! × k₂! × … × kₘ!)
Result: n! shown by the calculator
Using the preset "C(5,2)", the calculator evaluates the binomial coefficient setup, applies the selected algebra rules, and reports n! with supporting checks so you can verify each transformation.
This calculator takes n (total items), k (items chosen), k₂ (optional, for multinomial), k₃ (optional, for multinomial) and applies the relevant binomial coefficient relationships from your chosen method. It returns both final and intermediate values so you can audit the process instead of treating it as a black box.
Start with the primary output, then use n!, k!, (n−k)!, Multinomial Coefficient to confirm signs, magnitude, and internal consistency. If anything looks off, change one input and compare the updated outputs to isolate the issue quickly.
A strong workflow is manual solve first, calculator verify second. Repeating that loop improves speed and accuracy because you learn to spot common setup errors before they cost points on multi-step algebra problems.
Combinations C(n,k) count selections where order does not matter. Permutations P(n,k) count arrangements where order does matter. P(n,k) = C(n,k) × k!.
Pascal's triangle is a triangular array where each entry is the sum of the two entries above it. Row n of the triangle contains the binomial coefficients C(n,0), C(n,1), …, C(n,n).
A multinomial coefficient extends the binomial to more than two groups. It equals n! / (k₁! × k₂! × … × kₘ!) and counts ways to split n items into groups of sizes k₁, k₂, …, kₘ.
In the standard combinatorial definition, both n and k must be non-negative integers with k ≤ n. Generalized binomial coefficients allow real or negative n, but this calculator uses the standard definition.
There is exactly one way to choose zero items from any set — by choosing nothing. Mathematically, 0! = 1 by definition, so C(n,0) = n! / (0! × n!) = 1.
JavaScript can safely represent integers up to 2^53 − 1. For very large n, the factorial values overflow, but the calculator handles moderate values (n up to ~170) accurately.