Calculator for binomial theorem and binomial coefficients

Calculator for binomial coefficients:

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Calculator for binomial theorem:

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Binomial Coefficient and Binomial theorem

The defined for all natural numbers function

$\left(\begin{array}{c}n\\ k\end{array}\right)$ is called a binomial coefficient.

The binomial coefficient is defined as follows:

$$\left(\begin{array}{c}n\\ k\end{array}\right)=\frac{\mathrm{n!}}{\mathrm{k!}\left(n-k\right)!}\text{with}0\le k\le n\text{in other case}\left(\begin{array}{c}n\\ k\end{array}\right)=0$$

Properties of the binomial coefficients

Symmetry rule:

$\left(\begin{array}{c}n\\ k\end{array}\right)=\left(\begin{array}{c}n\\ n-k\end{array}\right)$

Addition rule:

$\left(\begin{array}{c}n\\ k\end{array}\right)+\left(\begin{array}{c}n\\ k+1\end{array}\right)=\left(\begin{array}{c}n+1\\ k+1\end{array}\right)$

Binomial theorem

For all real numbers a and b and all natural numbers n holds the binomial theorem:

${\left(a+b\right)}^n=\sum _{k=0}^n\left(\begin{array}{c}n\\ k\end{array}\right)a^{n-k}{b}^k=\left(\begin{array}{c}n\\ 0\end{array}\right)a^n{b}^0+\left(\begin{array}{c}n\\ 1\end{array}\right)a^{n-1}{b}^1+...+\left(\begin{array}{c}n\\ \mathrm{n}\end{array}\right)a^0{b}^n$

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