Coëfficiens binomialis
Numerus coëfficiens binomialis seu binominalis sic definitur: ( n k ) = n 1 ⋅ n − 1 2 ⋯ n − ( k − 1 ) k = n ⋅ ( n − 1 ) ⋯ ( n − k + 1 ) k ! = ∏ j = 1 k n + 1 − j j , {\displaystyle {\begin{aligned}{\binom {n}{k}}&={\frac {n}{1}}\cdot {\frac {n-1}{2}}\cdots {\frac {n-(k-1)}{k}}\\&={\frac {n\cdot (n-1)\cdots (n-k+1)}{k!}}\\&=\prod _{j=1}^{k}{\frac {n+1-j}{j}},\end{aligned}}} ubi k ! {\displaystyle {k!}} est factorialis numeri integri k {\displaystyle {k}} . Coëfficientes binomiales sunt partes theorematis binomialis: ( x + y ) n = ( n 0 ) x n + ( n 1 ) x n − 1 y + … + ( n n − 1 ) x y n − 1 + ( n n ) y n = ∑ k = 0 n ( n k ) x n − k y k . {\displaystyle (x+y)^{n}={\binom {n}{0}}x^{n}+{\binom {n}{1}}x^{n-1}y+\ldots +{\binom {n}{n-1}}xy^{n-1}+{\binom {n}{n}}y^{n}=\sum _{k=0}^{n}{\binom {n}{k}}x^{n-k}y^{k}.} Coëfficientes binomiales perutilia sunt in arte combinatoria et Triangulum arithmeticum Pascalianum.