# 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:

$( n k ) = n! k! ( n - k ) ! with0≤k≤n in other case ( n k ) = 0$

Properties of the binomial coefficients

Symmetry rule:

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

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

Binomial theorem

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

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

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