Problems from Combinatorial numbers

Calculate the value of the following combinatorial numbers:

  1. $$ \begin{pmatrix} 8 \\ 3 \end{pmatrix}$$
  2. $$ \begin{pmatrix} 4 \\ 1 \end{pmatrix}$$
  3. $$ \begin{pmatrix} 7 \\ 2 \end{pmatrix}$$
  4. $$ \begin{pmatrix} 157 \\ 0 \end{pmatrix}$$
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Development:

  1. $$ \begin{pmatrix} 8 \\ 3 \end{pmatrix}=\dfrac{8!}{3!\cdot(8-3)!}= \dfrac{8\cdot7\cdot6\cdot5!}{3!\cdot5!}=\dfrac{8\cdot7\cdot6}{3\cdot2}=56 $$

  2. $$ \begin{pmatrix} 4 \\ 1 \end{pmatrix}=4 $$

  3. $$ \begin{pmatrix} 7 \\ 2 \end{pmatrix}=\dfrac{7!}{2!\cdot(7-2)!}= \dfrac{7\cdot6\cdot5!}{2!\cdot5!}=7\cdot3=21 $$

  4. $$ \begin{pmatrix} 157 \\ 0 \end{pmatrix}=1$$

Solution:

  1. $$ 56 $$

  2. $$ 4 $$

  3. $$ 21 $$

  4. $$ 1$$
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