This online binomial coefficient calculator computes the value of a binomial coefficient C(n,k) given values of the parameters n and k, that must be non-negative integers in the range of 0 ≤ k ≤ n < 1030. In case of k << n the parameter n can significantly exceed the above mentioned upper threshold.
C(n,k) = n! / k!(n-k)!
Binomial coefficient formula
The binomial coefficient is defined as the number of different ways to choose a \(k\)-element subset from an \(n\)-element set. In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers \(n ≥ k ≥ 0\) and can be expressed in the form:
$${ C }_{ n }^{ k }=\frac { n! }{ k!(n-k)! } .$$
Binomial coefficients have many different properties. The simplest of them are easily deducted from the above formula.
• Symmetry rule:
$${ C }_{ n }^{ k }={ C }_{ n }^{ n-k }$$
• Recurrence relation:
$${ C }_{ n }^{ k }={ C }_{ n-1 }^{ k-1 }+{ C }_{ n-1 }^{ k }.$$
These formulas are used in this online binomial coefficient calculator to calculate binomial coefficients.
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