Binomial Coefficient Calculator
The binomial coefficient C(n, r), read as "n choose r," counts the number of distinct subsets of size r that can be formed from a set of n elements. It is the cornerstone of combinatorics and appears in the binomial theorem, Pascal's triangle, probability theory, and the calculation of combinations. The formula is C(n, r) = n! / (r! * (n-r)!), but this calculator uses an iterative multiplication method to avoid computing large factorials directly, keeping results accurate for practical n values. Enter n (total items) and r (items to choose) to compute the result.
Binomial coefficient formula
C(n, r) = n! / (r! * (n - r)!)
Computed iteratively as the product of (n - r + 1) * (n - r + 2) * ... * n divided by r! to avoid overflow. This equals the r-th entry in row n of Pascal's triangle.
Key properties
- C(n, r) = C(n, n - r): choosing r items is equivalent to choosing which n - r to exclude.
- C(n, 0) = C(n, n) = 1 for all n.
- Pascal's identity: C(n, r) = C(n-1, r-1) + C(n-1, r).
- The sum of all C(n, r) for r from 0 to n equals 2^n.
- C(n, r) is always a positive integer when 0 <= r <= n.
Binomial coefficient: frequently asked questions
What is a binomial coefficient?
The binomial coefficient C(n, r) counts the number of ways to choose r items from a set of n items without regard to order. It equals n! / (r! * (n-r)!) and appears as the coefficients in the expansion of (x + y)^n.
How is C(n, r) different from a permutation?
A permutation P(n, r) counts ordered selections: P(n, r) = n! / (n-r)!. A combination C(n, r) divides by r! to remove duplicates from order, so C(n, r) = P(n, r) / r!. Combinations count selections where order does not matter.
What is C(n, 0) and C(n, n)?
C(n, 0) = 1 for any n because there is exactly one way to choose zero items (choose nothing). C(n, n) = 1 because there is exactly one way to choose all n items. These are the boundary cases of Pascal's triangle.
How do binomial coefficients appear in probability?
In binomial probability, P(exactly k successes in n trials) = C(n, k) * p^k * (1-p)^(n-k). The coefficient C(n, k) counts the number of distinct ways to arrange k successes among n trials.
What is Pascal's identity?
Pascal's identity states that C(n, r) = C(n-1, r-1) + C(n-1, r). Each entry in Pascal's triangle equals the sum of the two entries directly above it. This recursive structure allows efficient computation of all binomial coefficients.
Official sources
- NIST Digital Library of Mathematical Functions, Section 26.3: dlmf.nist.gov/26.3.
- NIST, Mathematical topics: nist.gov/topics/mathematics.
Reviewed by the CalculatorHub team, edited by James Graham, 15 June 2026. See our methodology.