Binomial coefficient

Number of subsets of a given size

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 is written ( n k) {\displaystyle {\tbinom {n}{k}}} or ⁠ C ( n , k) {\displaystyle C(n,k)} ⁠. It is the coefficient of the xk term in the polynomial expansion of the binomial power (1 + x)n; this coefficient can be computed by the multiplicative formula ( n k) = n × ( n − 1) × ⋯ × ( n − k + 1) k × ( k − 1) × ⋯ × 1 , {\displaystyle {\binom {n}{k}}={\frac {n\times (n-1)\times \cdots \times (n-k+1)}{k\times (k-1)\times \cdots \times 1}},} which using factorial notation can be compactly expressed as ( n k) = n !

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