Решебник по алгебре и начала математического анализа 10 класс Алимов Задание 1076

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\[1)\ \frac{A_{15}^{9} - A_{15}^{8}}{A_{15}^{7}} =\]

\[= \frac{\frac{15!}{(15 - 9)!} - \frac{15!}{(15 - 8)!}}{\frac{15!}{(15 - 7)!}} =\]

\[= \frac{\frac{15!}{6!} - \frac{15!}{7!}}{\frac{15!}{8!}} = \frac{8!}{6!} - \frac{8!}{7!} =\]

\[= \frac{8 \bullet 7 \bullet 6!}{6!} - \frac{8 \bullet 7!}{7!} = 8 \bullet 7 - 8 =\]

\[= 8 \bullet 6 = 48.\]

\[2)\ \frac{A_{18}^{10} + A_{18}^{11}}{A_{18}^{9}} =\]

\[= \frac{\frac{18!}{(18 - 10)!} + \frac{18!}{(18 - 11)!}}{\frac{18!}{(18 - 9)!}} =\]

\[= \frac{\frac{18!}{8!} + \frac{18!}{7!}}{\frac{18!}{9!}} = \frac{9!}{8!} + \frac{9!}{7!} =\]

\[= \frac{9 \bullet 8!}{8!} + \frac{9 \bullet 8 \bullet 7!}{7!} = 9 + 9 \bullet 8 =\]

\[= 9 \bullet 9 = 81.\]

\[3)\ \frac{A_{9}^{4} \bullet A_{4}^{4}}{A_{8}^{6}} =\]

\[= \frac{9 \bullet 8 \bullet 7 \bullet 6 \bullet 4 \bullet 3 \bullet 2 \bullet 1}{8 \bullet 7 \bullet 6 \bullet 5 \bullet 4 \bullet 3} =\]

\[= \frac{9 \bullet 2}{5} = \frac{18}{5}.\]

\[4)\ \frac{A_{5}^{5} \bullet A_{10}^{3}}{A_{9}^{7}} =\]

\[= \frac{5 \bullet 4 \bullet 3 \bullet 2 \bullet 1 \bullet 10 \bullet 9 \bullet 8}{9 \bullet 8 \bullet 7 \bullet 6 \bullet 5 \bullet 4 \bullet 3} =\]

\[= \frac{2 \bullet 10}{7 \bullet 6} = \frac{20}{42} = \frac{10}{21}.\]