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МатематикаРешебник по алгебре и начала математического анализа 11 класс Алимов Задание 1109
Задание 1109
\[\boxed{\mathbf{1109}\mathbf{.}}\]
\[1)\ C_{4}^{1} + C_{4}^{2} + C_{4}^{3} + C_{4}^{4} =\]
\[= 2^{4} - C_{4}^{0} = 16 - 1 = 15\]
\[2)\ C_{5}^{1} + C_{5}^{2} + C_{5}^{3} + C_{5}^{4} =\]
\[= 2^{5} - C_{5}^{0} - C_{5}^{5} = 32 - 1 - 1 =\]
\[= 30\]
\[5)\ C_{12}^{2} + C_{12}^{3} + C_{13}^{4} + C_{14}^{5} =\]
\[= C_{13}^{3} + C_{13}^{4} + C_{14}^{5} = C_{14}^{4} + C_{14}^{5} =\]
\[= C_{15}^{5}\]
\[C_{15}^{5} = \frac{15!}{(15 - 5)! \bullet 5!} = \frac{15!}{10! \bullet 5!} =\]
\[= \frac{15 \bullet 14 \bullet 13 \bullet 12 \bullet 11 \bullet 10}{10! \bullet 5 \bullet 4 \bullet 3 \bullet 2} =\]
\[= 3 \bullet 7 \bullet 13 \bullet 11 = 3003\]
\[6)\ C_{9}^{3} + C_{9}^{4} + C_{10}^{5} + C_{11}^{6} =\]
\[= C_{10}^{4} + C_{10}^{5} + C_{11}^{6} = C_{11}^{5} + C_{11}^{6} =\]
\[= C_{12}^{6}\]
\[C_{12}^{6} = \frac{12!}{(12 - 6)! \bullet 6!} = \frac{12!}{6! \bullet 6!} =\]
\[= \frac{12 \bullet 11 \bullet 10 \bullet 9 \bullet 8 \bullet 7 \bullet 6!}{6 \bullet 5 \bullet 4 \bullet 3 \bullet 2 \bullet 6!} =\]
\[= 2 \bullet 11 \bullet 2 \bullet 3 \bullet 7 = 924\]
\[7)\ C_{21}^{7} - C_{20}^{7} = C_{20}^{6}\]
\[C_{20}^{6} = \frac{20!}{(20 - 6)! \bullet 6!} = \frac{20!}{14! \bullet 6!} =\]
\[= \frac{20 \bullet 19 \bullet 18 \bullet 17 \bullet 16 \bullet 15 \bullet 14!}{14! \bullet 6 \bullet 5 \bullet 4 \bullet 3 \bullet 2} =\]
\[= 10 \bullet 19 \bullet 3 \bullet 17 \bullet 4 = 38\ 760\]
\[8)\ C_{25}^{9} - C_{24}^{9} = C_{24}^{8}\]
\[C_{24}^{8} = \frac{24!}{(24 - 8)! \bullet 8!} = \frac{24!}{16! \bullet 8!} =\]
\[= 3 \bullet 23 \bullet 11 \bullet 3 \bullet 19 \bullet 17 =\]
\[= 735\ 471\]
\[9)\ C_{15}^{10} - C_{14}^{9} = C_{14}^{10}\]
\[C_{14}^{10} = \frac{14!}{(14 - 10)! \bullet 10!} = \frac{14!}{4! \bullet 10!} =\]
\[= \frac{14 \bullet 13 \bullet 12 \bullet 11 \bullet 10!}{4 \bullet 3 \bullet 2 \bullet 10!} =\]
\[= 7 \bullet 13 \bullet 11 = 1001\]
\[10)\ C_{13}^{8} - C_{12}^{7} = C_{12}^{8}\]
\[C_{12}^{8} = \frac{12!}{(12 - 8)! \bullet 8!} = \frac{12!}{4! \bullet 8!} =\]
\[= \frac{12 \bullet 11 \bullet 10 \bullet 9 \bullet 8!}{4 \bullet 3 \bullet 2 \bullet 8!} =\]
\[= 3 \bullet 11 \bullet 5 \bullet 3 = 495\]
