The Binomial Coefficient Calculator provided by Hesapstan computes C(n,k), or n choose k, exactly as an integer for 0 ≤ k ≤ n ≤ 1000. It does not calculate binomial probability P(X=k); it only returns the binomial coefficient used in combinations, Pascal's triangle, and the binomial theorem.
C(n,k) counts unordered selections from n items
The binomial coefficient C(n,k), also read as n choose k, gives the number of ways to choose k items from n distinct items when order does not matter. The same value appears as a coefficient in the binomial theorem and as an entry in Pascal's triangle.
C(n,k) is the combination count. This page computes the coefficient itself; it does not include a probability p, a random variable X, or a binomial distribution calculation.
The formula can be read as factorial form or reduced product form
The standard formula is C(n,k) = n! / (k! × (n−k)!). The calculator also shows the reduced product form: (n × (n−1) × … × (n−k+1)) / k!. This form is more practical for large n because it avoids expanding unnecessary factorial terms.
The runtime uses BigInt arithmetic, so large values such as C(1000,500) are returned as exact integer strings instead of rounded floating-point or scientific-notation approximations.
Symmetry reduces the work without changing the value
Binomial coefficients satisfy C(n,k) = C(n,n−k). For example, C(20,18) is the same as C(20,2), and both equal 190. When k is larger than n−k, the calculator uses this symmetry and shows a note to make the shortcut visible.
n and k must be non-negative integers, k must not exceed n, and n must be at most 1000. Negative values, decimals, and k > n are rejected.
Pascal's triangle is shown only for small n
The nth row of Pascal's triangle is C(n,0), C(n,1), …, C(n,n). For n ≤ 12, the calculator displays that row as a visual cross-check. For larger n, the row is not shown because it becomes too wide for a useful interface.
C(52,5) = 2,598,960. This is the number of unordered 5-item selections from 52 items; it is not a probability by itself.
Worked example: C(10,3) = 120
To find the number of unordered 3-item selections from 10 distinct items, calculate C(10,3).
- Factorial formula: C(10,3) = 10! / (3! × 7!)
- Reduced product form: (10 × 9 × 8) / (3 × 2 × 1)
- Numerator: 10 × 9 × 8 = 720
- Denominator: 3! = 6
- Result: 720 / 6 = 120
Instead of computing C(20,18) directly, apply symmetry: C(20,18) = C(20,20−18) = C(20,2) = (20×19) / (2×1) = 190. For large k, this shortcut simplifies the calculation significantly.
Binomial coefficients appear as coefficients in the binomial expansion
In the expansion of (a+b)^n, the coefficient of the term a^k · b^(n−k) is exactly C(n,k). For example, expanding (a+b)^3 gives coefficients C(3,0)=1, C(3,1)=3, C(3,2)=3, C(3,3)=1. These are the same as the third row of Pascal's triangle.
It returns only the value C(n,k). Algebraic expansions of (a+b)^n and binomial probability calculations P(X=k) = C(n,k)p^k(1−p)^(n−k) are outside this page's scope.
Frequently Asked Questions
Does this calculator compute binomial probability?
No. It computes only C(n,k). Binomial probability also requires p^k and (1−p)^(n−k), which are not part of this runtime.
Is a binomial coefficient the same as a combination?
The value C(n,k) is the combination count. In counting it means choosing k from n; in algebra it appears as a coefficient in the binomial theorem.
What range of n and k is supported?
The calculator accepts integers with 0 ≤ k ≤ n ≤ 1000. Negative values, decimal values, and k > n are blocked by validation.
Why is the result exact?
The runtime uses BigInt integer arithmetic, so the result is returned exactly as an integer string without floating-point rounding.
Why are C(n,0) and C(n,n) equal to 1?
There is exactly one way to choose nothing from n items, and exactly one way to choose all n items.
What does the symmetry note mean?
It means the calculator used C(n,k) = C(n,n−k) to compute the same value with the smaller side of the selection.
Why is the Pascal row not always shown?
Pascal rows become wide as n grows. The runtime displays the row only for n ≤ 12.
Does this support multinomial coefficients?
No. Multinomial coefficients and multiple group selections are outside the scope of this calculator.