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.

## Bibliography¶

“Binomial Coefficient.” 2023. Wikipedia, January. https://en.wikipedia.org/w/index.php?title=Binomial_coefficient&oldid=1133587426.