Question

$\\\frac{ A_{10}^{6} - A_{10}^{5}}{A_{9}^{5}-A_{9}^{4}}$
Вычислите

Answer 1

Работая с факториалами:

$\dfrac{ A_{10}^{6} - A_{10}^{5}}{A_{9}^{5}-A_{9}^{4}}=\dfrac{ \dfrac{10!}{(10-6)!} - \dfrac{10!}{(10-5)!}}{\dfrac{9!}{(9-5)!}-\dfrac{9!}{(9-4)!}}=\dfrac{ \dfrac{10!}{4!} - \dfrac{10!}{5!}}{\dfrac{9!}{4!}-\dfrac{9!}{5!}}=$

$=\dfrac{\dfrac{9!}{4!}\cdot\left(10-\dfrac{10}{5}\right)}{ \dfrac{9!}{4!}\cdot\left(1-\dfrac{1}{5}\right)}=\dfrac{ 10- \dfrac{10}{5}}{ 1-\dfrac{1}{5}}=\dfrac{10-2}{1-0.2}=\dfrac{8}{0.8}=10$

Или расписывая факториалы:

$\dfrac{ A_{10}^{6} - A_{10}^{5}}{A_{9}^{5}-A_{9}^{4}}=\dfrac{ \dfrac{10!}{(10-6)!} - \dfrac{10!}{(10-5)!}}{\dfrac{9!}{(9-5)!}-\dfrac{9!}{(9-4)!}}=\dfrac{10\cdot9\cdot8\cdot7\cdot6\cdot5-10\cdot9\cdot8\cdot7\cdot6}{9\cdot8\cdot7\cdot6\cdot5-9\cdot8\cdot7\cdot6} =$

$=\dfrac{9\cdot8\cdot7\cdot6\cdot(10\cdot5-10)}{9\cdot8\cdot7\cdot6\cdot(5-1)} =\dfrac{10\cdot5-10}{5-1} =\dfrac{50-10}{4} =10$