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МатематикаРешебник по алгебре и начала математического анализа 11 класс Колягин Проверь себя V. Сложный вариант
Проверь себя V. Сложный вариант
\[\mathbf{1}\text{.\ }\]
\[1)\ P_{n + 4} = 20P_{n + 2}\]
\[(n + 4)! = 20(n + 2)!\]
\[(n + 4)(n + 3)(n + 2)! =\]
\[= 20(n + 2)!\]
\[n^{2} + 3n + 4n + 12 = 20\]
\[n^{2} + 7n - 8 = 0\]
\[D = 49 + 32 = 81\]
\[n_{1} = \frac{- 7 - 9}{2} = - 8;\]
\[n_{2} = \frac{- 7 + 9}{2} = 1.\]
\[Ответ:\ \ 1.\]
\[2)\ A_{n + 1}^{3} = C_{n + 2}^{2}\]
\[\frac{(n + 1)!}{(n - 2)!} = \frac{(n + 2)!}{2! \bullet n!}\]
\[\frac{(n + 1)!}{(n - 2)!} = \frac{(n + 2)(n + 1)!}{2n(n - 1)(n - 2)!}\]
\[2n^{2} - 2n = n + 2\]
\[2n^{2} - 3n - 2 = 0\]
\[D = 9 + 16 = 25\]
\[n_{1} = \frac{3 - 5}{2 \bullet 2} = - \frac{1}{2};\]
\[n_{2} = \frac{3 + 5}{2 \bullet 2} = 2.\]
\[Ответ:\ \ 2.\]
\[\mathbf{2}\text{.\ }\]
\[(n - 3)P_{n + 2} < P_{n + 1}\]
\[(n - 3)(n + 2)! < (n + 1)!\]
\[(n - 3)(n + 2)(n + 1)! < (n + 1)!\]
\[n^{2} - 3n + 2n - 6 < 1\]
\[n^{2} - n - 7 < 0\]
\[D = 1 + 28 = 29\]
\[n = \frac{1 \pm \sqrt{29}}{2};\]
\[- 3 < \frac{1 - \sqrt{29}}{2} < n < \frac{1 + \sqrt{29}}{2} < 4.\]
\[Ответ:\ \ 0;\ 1;\ 2;\ 3.\]
\[\mathbf{3.}\ \]
\[Трехзначных\ чисел:\]
\[n = \left\{ 1;\ 2;\ 3;\ 4;\ 5 \right\};\]
\[N_{1} = 5 \bullet 4 \bullet 3 = 60.\]
\[Пятизначных\ чисел:\]
\[n = \left\{ 6;\ 7;\ 8 \right\};\]
\[N_{2} = 3 \bullet 3 \bullet 3 \bullet 3 \bullet 3 = 243.\]
\[Общее\ число\ пар:\]
\[N = N_{1} \bullet N_{2} = 60 \bullet 243 = 14\ 580.\]
\[Ответ:\ \ 14\ 580.\]
\[\mathbf{4}.\]
\[N = C_{6}^{0} + C_{6}^{1} + C_{6}^{2} + C_{6}^{3} + C_{6}^{4} + C_{6}^{5} + C_{6}^{6} =\]
\[= 2^{6} = 64.\]
\[Ответ:\ \ 64.\]
\[\mathbf{5}\text{.\ }\]
\[1)\ C_{10}^{7} - C_{9}^{6} = C_{9}^{7} + C_{9}^{6} - C_{9}^{6} =\]
\[= C_{9}^{7} = \ C_{9}^{2} = \frac{9!}{7! \bullet 2!} = \frac{9 \bullet 8 \bullet 7!}{7! \bullet 2} =\]
\[= 9 \bullet 4 = 36.\]
\[Ответ:\ \ 36.\]
\[2)\ C_{9}^{5} + C_{9}^{6} + C_{9}^{7} + C_{9}^{8} + C_{9}^{9} =\]
\[= 2^{9} - C_{9}^{0} - C_{9}^{1} - C_{9}^{2} - C_{9}^{3} - C_{9}^{4} =\]
\[= 2^{9} - \left( C_{9}^{5} + C_{9}^{6} + C_{9}^{7} + C_{9}^{8} + C_{9}^{9} \right) =\]
\[= 2^{9} \bullet \frac{1}{2} = 2^{8} = 256.\]
\[Ответ:\ \ 256.\]
\[\mathbf{6}\text{.\ }\]
\[\left( \sqrt{x} - \frac{1}{\sqrt[3]{x}} \right)^{20}\]
\[C_{20}^{n} \bullet \left( \sqrt{x} \right)^{20 - n} \bullet \left( \frac{1}{\sqrt[3]{x}} \right)^{n} = C_{20}^{n} \bullet x^{5}\]
\[x^{10 - \frac{1}{2}n - \frac{1}{3}n} = x^{5}\]
\[10 - \frac{5}{6}n = 5\ \]
\[\frac{5}{6}n = 5\]
\[n = 6.\]
\[N = C_{20}^{6} \bullet x^{5} = \frac{20!}{14! \bullet 6!} \bullet x^{5} =\]
\[= \frac{20 \bullet 19 \bullet 18 \bullet 17 \bullet 16 \bullet 15}{6 \bullet 5 \bullet 4 \bullet 3 \bullet 2} \bullet x^{5} =\]
\[= 38\ 760 \bullet x^{5}.\]
\[Ответ:\ \ 38\ 760 \bullet x^{5}.\]
