Question

Sa se arate că numărul 10!•9! Divine 20!

Answer 1

$\dfrac{20!}{10!\cdot 9!} = \dfrac{10! \cdot 11\cdot 12\cdot 13\cdot ...\cdot 20}{10! \cdot 9!} = \dfrac{11\cdot 12\cdot 13\cdot ... \cdot 20}{9!} = \\ \\ = \dfrac{11\cdot (\bold{2}\cdot \bold{6})\cdot 13\cdot (2\cdot \bold{7})\cdot (\bold{3}\cdot \bold{5})\cdot (2\cdot \bold{8}) \cdot 17\cdot (2\cdot \bold{9})\cdot 19\cdot (\bold{4}\cdot 5)}{2\cdot 3\cdot 4\cdot 5\cdot 6\cdot 7\cdot 8\cdot 9} = \\ \\ = 11\cdot 13\cdot 2\cdot 2\cdot 17\cdot 2 \cdot 19\cdot 5 \in \mathbb{N} \\ \\ \Rightarrow (10!\cdot 9!) ~|~ 20!$