Calculate the value of the following combinatorial numbers:
- $$ \begin{pmatrix} 8 \\ 3 \end{pmatrix}$$
- $$ \begin{pmatrix} 4 \\ 1 \end{pmatrix}$$
- $$ \begin{pmatrix} 7 \\ 2 \end{pmatrix}$$
- $$ \begin{pmatrix} 157 \\ 0 \end{pmatrix}$$
See development and solution
Development:
-
$$ \begin{pmatrix} 8 \\ 3 \end{pmatrix}=\dfrac{8!}{3!\cdot(8-3)!}= \dfrac{8\cdot7\cdot6\cdot5!}{3!\cdot5!}=\dfrac{8\cdot7\cdot6}{3\cdot2}=56 $$
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$$ \begin{pmatrix} 4 \\ 1 \end{pmatrix}=4 $$
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$$ \begin{pmatrix} 7 \\ 2 \end{pmatrix}=\dfrac{7!}{2!\cdot(7-2)!}= \dfrac{7\cdot6\cdot5!}{2!\cdot5!}=7\cdot3=21 $$
- $$ \begin{pmatrix} 157 \\ 0 \end{pmatrix}=1$$
Solution:
-
$$ 56 $$
-
$$ 4 $$
-
$$ 21 $$
- $$ 1$$