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Mathematics-Online course: Prepcourse Mathematics - Basics - Combinatorics

Pascal's Triangle

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The binomial coefficients

$\displaystyle \binom{n}{k} = \frac{n!}{(n-k)!\,k!} $

can be recursively calculated by

$\displaystyle \binom{n+1}{k} = \binom{n}{k-1} + \binom{n}{k} \,. $

The recursion, known as Pascal's triangle, is illustrated in the figure.

\begin{center} \begin{tabular}{c\vert cccccccccccccc} $\displaystyle \binom{0}{k... ...dots$\ & & $\vdots$\ & & $\vdots$\ & & $\vdots$\ & \\ \end{tabular}\end{center}

(Authors: Höllig/Knesch/Abele)
The assertion

$\displaystyle \frac{(n+1)!}{(n+1-k)!\,k!} = \frac{n!}{(n-k+1)!\,(k-1)!} + \frac{n!}{(n-k)!\,k!} $

reduces to the trivial identity

$\displaystyle n+1 = k + (n+1-k) $

if we divide both sides by $ n!$ and multiply by $ (n-k+1)!\,k!$.
(Authors: Höllig/Knesch/Abele)
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  automatically generated 10/23/2009