Symbol Newtona: $\left(\begin{array}{c}n\\ k\end{array}\right)=\frac{n!}{k!\cdot \left(n-k\right)!},\text{gdzie}k,n\in \mathbb{N}\text{i}k\le n$

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a)

$\left(\begin{array}{c}7\\ 3\end{array}\right)=\frac{7!}{3!\cdot \left(7-3\right)!}=\frac{7!}{3!\cdot 4!}=\frac{7\cdot 6\cdot 5\cdot 4!}{1\cdot 2\cdot 3\cdot 4!}=\frac{7\cdot 6\cdot 5}{1\cdot 2\cdot 3}=\frac{210}{6}=35$  

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b) 

$\left(\begin{array}{c}20\\ 2\end{array}\right)=\frac{20!}{2!\cdot \left(20-2\right)!}=\frac{20!}{2!\cdot 18!}=\frac{20\cdot 19\cdot 18!}{1\cdot 2\cdot 18!}=\frac{20\cdot 19}{1\cdot 2}=\frac{380}{2}=190$  

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c)

$\left(\begin{array}{c}17\\ 1\end{array}\right)=\frac{17!}{1!\cdot \left(17-1\right)!}=\frac{17!}{1!\cdot 16!}=\frac{17\cdot 16!}{1\cdot 16!}=\frac{17}{1}=17$  

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d)

$\left(\begin{array}{c}13\\ 13\end{array}\right)=\frac{13!}{13!\cdot \left(13-13\right)!}=\frac{13!}{13!\cdot 0!}=\frac{1}{1}=1$  

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e)

$\left(\begin{array}{c}8\\ 0\end{array}\right)=\frac{8!}{0!\cdot \left(8-0\right)!}=\frac{8!}{0!\cdot 8!}=\frac{1}{1}=1$