Pascal's triangle

Definition

Let $n,k\in \mathbb {N}$, then the entry in the n-th row and k-th column of Pascal's triangle is denoted with:

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

which fulfills the rule:

${\binom {n}{k}}={\binom {n-1}{k-1}}+{\binom {n-1}{k}}$

for all $k,n\geq 1$. With those binomial coefficients the triangle can be visualized as follows:

${\begin{array}{ccccccccc}&&&&\displaystyle {\binom {0}{0}}&&&&\\&&&&&&&&\\&&&\displaystyle {\binom {1}{0}}&&\displaystyle {\binom {1}{1}}&&&\\&&&&&&&&\\&&\displaystyle {\binom {2}{0}}&&\displaystyle {\binom {2}{1}}&&\displaystyle {\binom {2}{2}}&&\\&&&&&&&&\\&\displaystyle {\binom {3}{0}}&&\displaystyle {\binom {3}{1}}&&\displaystyle {\binom {3}{2}}&&\displaystyle {\binom {3}{3}}&\\&&&&&&&&\\\dots &&\dots &&\dots &&\dots &&\dots \\\end{array}}$

which continues with $n=4,5,\dots$; equivalent to the binomial coefficients are their values, which are as follows:

${\begin{array}{ccccccccc}&&&&1&&&&\\&&&&&&&&\\&&&1&&1&&&\\&&&&&&&&\\&&1&&2&&1&&\\&&&&&&&&\\&1&&3&&3&&1&\\&&&&&&&&\\\dots &&\dots &&\dots &&\dots &&\dots \\\end{array}}$

Those entries continue as well.

Corollary

Let $n\in \mathbb {N}$, then we have:

$\displaystyle \sum \limits _{k=0}^{n}~\displaystyle {\binom {n}{k}}=2^{n}$

Proof

We use the binomial theorem:

Fehler beim Parsen (Syntaxfehler): {\displaystyle 2^n = (1 + 1)^n = \displaystyle\sum\limits_{k = 0}^{n}~\displaystyle\binom{n}{k} 1^k 1^{n - k} = \displaystyle\sum\limits_{k = 0}^{n}~\displaystyle\binom{n}{k} \\ }

The series describes a row in the triangle of Pascal.

Remark

Pascal's triangle has many applications and is fundamental to combinatorics. In combinatorial number theory it takes a pivotal role as it describes the distribution of the (natural appearing) combinatorial trigonometric series of the second kind.

Symmetry

We point out, that Pascal's triangle has a symmetry:

Fehler beim Parsen (SVG (MathML kann über ein Browser-Plugin aktiviert werden): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle \displaystyle\binom{n}{k} = \displaystyle\binom{n}{n - k} \\}

that is not always preserved throughout the other binomial triangles.

Distribution

Datei:Different distributions in comparison.png

Different distributions

From the established entries of the positive binomial coefficients of the Pascal's triangle follows, that underlying it has a binomial distribution.