De driehoek van Pascal en het binomium van Newton

Wat is het verband tussen de driehoek van Pascal en het binomium van Newton?

maarte
Leerling bovenbouw havo-vwo - woensdag 19 november 2003

Antwoord

Driehoek van Pascal

                             1
                           1   1
                         1   2   1
                       1   3   3   1
                     1   4   6   4   1
                   1   5  10   10  5   1
                 1   6  15  20   15  6   1
               1   7  21  35   35  21  7   1

Binomium van Newton

(a+b)⁰ = 1
(a+b)¹ = 1.a+1.b
(a+b)² = 1.a² + 2.ab + 1.b²
(a+b)³ = 1.a³ + 3.a²b + 3.ab² + 1.b³
(a+b)⁴ = 1.a⁴ + 4.a³b + 6.a²b² + 4.ab³ + b⁴
(a+b)⁵ = 1.a⁵ + 5.a⁴b + 10.a³b² + 10.a²b³ + 5.ab⁴ + 1.b⁵
(a+b)⁶ = 1.a⁶ + 6.a⁵b + 15.a⁴b² + 20.a³b³ + 15.a²b⁴ + 6.ab⁵ + 1.b⁶

Samengevat

$
\begin{array}{c}
\left( {\begin{array}{*{20}c}
0 \\
0 \\
\end{array}} \right) \\
\begin{array}{*{20}c}
{\left( {\begin{array}{*{20}c}
1 \\
0 \\
\end{array}} \right)} & {\left( {\begin{array}{*{20}c}
1 \\
1 \\
\end{array}} \right)} \\
\end{array} \\
\begin{array}{*{20}c}
{\left( {\begin{array}{*{20}c}
2 \\
0 \\
\end{array}} \right)} & {\left( {\begin{array}{*{20}c}
2 \\
1 \\
\end{array}} \right)} & {\left( {\begin{array}{*{20}c}
2 \\
2 \\
\end{array}} \right)} \\
\end{array} \\
\begin{array}{*{20}c}
{\left( {\begin{array}{*{20}c}
3 \\
0 \\
\end{array}} \right)} & {\left( {\begin{array}{*{20}c}
3 \\
1 \\
\end{array}} \right)} & {\left( {\begin{array}{*{20}c}
3 \\
2 \\
\end{array}} \right)} & {\left( {\begin{array}{*{20}c}
3 \\
3 \\
\end{array}} \right)} \\
\end{array} \\
\begin{array}{*{20}c}
{\left( {\begin{array}{*{20}c}
4 \\
0 \\
\end{array}} \right)} & {\left( {\begin{array}{*{20}c}
4 \\
1 \\
\end{array}} \right)} & {\left( {\begin{array}{*{20}c}
4 \\
2 \\
\end{array}} \right)} & {\left( {\begin{array}{*{20}c}
4 \\
3 \\
\end{array}} \right)} & {\left( {\begin{array}{*{20}c}
4 \\
4 \\
\end{array}} \right)} \\
\end{array} \\
\end{array}
$

Over de driehoek van Pascal kan je meer lezen op volgende links:

• Binomium van Newton en de driehoek van Pascal
• Driehoek van Pascal
• Driehoek van Pascal bij Wisfaq

Mvg

Els

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