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 \(\tbinom {n}{k}\). 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

\(\binom {n}{k}}={\frac {n\times (n-1)\times \cdots \times (n-k+1)}{k\times (k-1)\times \cdots \times 1}\),

which using factorial notation can be compactly expressed as

\(\binom {n}{k}}={\frac {n!}{k!(n-k)!}\).

For example, the fourth power of 1 + x is

\(\begin{aligned}(1+x)^{4}&={\tbinom {4}{0}}x^{0}+{\tbinom {4}{1}}x^{1}+{\tbinom {4}{2}}x^{2}+{\tbinom {4}{3}}x^{3}+{\tbinom {4}{4}}x^{4}\&=1+4x+6x^{2}+4x^{3}+x^{4},\end{aligned}\)

and the binomial coefficient \(\tbinom {4}{2}}={\tfrac {4\times 3}{2\times 1}}={\tfrac {4!}{2!2!}}=6\) is the coefficient of the \(x^2\) term.

(“Binomial Coefficient” 2023)



“Binomial Coefficient.” 2023. Wikipedia, January.