Упр.1076 ГДЗ Алимов 10-11 класс (Алгебра)

Найти значение выражения:

1. $$\frac{A_{15}^{9}-A_{15}^{8}}{A_{15}^{7}}$$
2. $$\frac{A_{18}^{10}-A_{18}^{11}}{A_{18}^{9}}$$
3. $$\frac{A_{9}^{4}\cdot A_{4}^{4}}{A_{8}^{6}}$$
4. $$\frac{A_{5}^{5}\cdot A_{10}^{3}}{A_{9}^{7}}$$

1) $$\frac{A_{15}^{9}-A_{15}^{8}}{A_{15}^{7}}=\frac{\frac{15!}{6!}-\frac{15!}{7!}}{\frac{15!}{8!}}=\frac{\frac{15!}{8!}\left(\frac{8!}{6!}-\frac{8!}{7!}\right)}{\frac{15!}{8!}}=\frac{8!}{6!}-\frac{8!}{7!}=8\cdot 7-8=48.$$

2) $$\frac{A_{18}^{10}+A_{18}^{11}}{A_{18}^{9}}=\frac{\frac{18!}{8!}+\frac{18!}{7!}}{\frac{18!}{9!}}=\frac{\frac{18!}{9!}\left(\frac{9!}{8!}+\frac{9!}{7!}\right)}{\frac{18!}{9!}}=\frac{9!}{8!}+\frac{9!}{7!}=9+9\cdot 8=81.$$

3) $$\frac{A_{9}^{4}\cdot A_{4}^{4}}{A_{8}^{6}}=\frac{\frac{9!}{5!}\cdot 4!}{\frac{8!}{2!}}=\frac{9\cdot 8\cdot 7\cdot 6\cdot 4\cdot 3\cdot 2\cdot 1}{8\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3}=\frac{9\cdot 2}{5}=\frac{18}{5}.$$

4) $$\frac{A_{5}^{5}\cdot A_{10}^{3}}{A_{9}^{7}}=\frac{5!\cdot \frac{10!}{7!}}{\frac{9!}{2!}}=\frac{5!\cdot 10\cdot 9\cdot 8}{9!\!/2}=\frac{2\cdot 10}{7\cdot 6}=\frac{20}{42}=\frac{10}{21}.$$

Ответ: 1) $$48$$; 2) $$81$$; 3) $$\frac{18}{5}$$; 4) $$\frac{10}{21}$$.