Coëfficiens binomialis

Dispositio coëfficientium binomialium in triangulo arithmetico Pascaliano, ubi quique numerus est summa duorum numerorum superiorum.

Illustratio coëfficientium binomialium

Numerus coëfficiens binomialis seu binominalis sic definitur:

$\begin{array}{cc}(\genfrac{}{}{0ex}{}{n}{k})& =\frac{n}{1}\cdot \frac{n-1}{2}\cdots \frac{n-(k-1)}{k}\\ & =\frac{n\cdot (n-1)\cdots (n-k+1)}{k!}\\ & =\prod _{j=1}^k\frac{n+1-j}{j},\end{array}$

ubi $k!$ est factorialis numeri integri $k$.

Coëfficientes binomiales sunt partes theorematis binomialis:

$(x+y{)}^n=(\genfrac{}{}{0ex}{}{n}{0})x^n+(\genfrac{}{}{0ex}{}{n}{1})x^{n-1}y+\dots +(\genfrac{}{}{0ex}{}{n}{n-1})x{y}^{n-1}+(\genfrac{}{}{0ex}{}{n}{n})y^n=\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})x^{n-k}{y}^k.$

Coëfficientes binomiales perutilia sunt in arte combinatoria et Triangulum arithmeticum Pascalianum.