Страница 335 номер 1109, ГДЗ по алгебре за 10-11 класс Алимов - ответы и решение
Найти значение выражения, предварительно его упростив: 1) \(C_4^1 + C_4^2 + C_4^3 + C_4^4\); 2) \(C_5^1 + C_5^2 + C_5^3 + C_5^4\); 3) \(C_7^0 + C_7^7 + C_7^1 + C_7^6 + C_7^2 + C_7^5 + C_7^3 + C_7^4\); 4) \(C_6^0 + C_6^6 + C_6^1 + C_6^5 + C_6^2 + C_6^4 + C_6^3\); 5) \(C_{12}^2 + C_{12}^3 + C_{13}^4 + C_{14}^5\); 6) \(C_9^3 + C_9^4 + C_{10}^5 + C_{11}^6\); 7) \(C_{21}^7 - C_{20}^7\); 8) \(C_{25}^9 - C_{24}^9\); 9) \(C_{15}^{10} - C_{14}^9\); 10) \(C_{13}^8 - C_{12}^7\).
1) \(C_{4}^{1} + C_{4}^{2} + C_{4}^{3} + C_{4}^{4} = 2^{4} - C_{4}^{0} = 16 - 1 = 15\);
2) \(C_{5}^{1} + C_{5}^{2} + C_{5}^{3} + C_{5}^{4} = 2^{5} - C_{5}^{0} - C_{5}^{5} = 32 - 2 = 30\);
3) \(C_{7}^{0} + C_{7}^{7} + C_{7}^{1} + C_{7}^{6} + C_{7}^{2} + C_{7}^{5} + C_{7}^{3} + C_{7}^{4} = 2^{7} = 128\);
4) \(C_{6}^{0} + C_{6}^{6} + C_{6}^{1} + C_{6}^{5} + C_{6}^{2} + C_{6}^{4} + C_{6}^{3} = 2^{6} = 64\);
5) \(C_{12}^{2} + C_{12}^{3} + C_{13}^{4} + C_{14}^{5} = C_{15}^{5} = \frac{15!}{5! \cdot 10!} = \frac{11 \cdot 12 \cdot 13 \cdot 14 \cdot 15}{120} = 3\ 003\);
6) \(C_{9}^{3} + C_{9}^{4} + C_{10}^{5} + C_{11}^{6} = C_{12}^{6} = \frac{12!}{6! \cdot 6!} = \frac{7 \cdot 8 \cdot 9 \cdot 10 \cdot 11 \cdot 12}{720} = 924\);
7) \(C_{21}^{7} - C_{20}^{7} = C_{20}^{6} + C_{20}^{7} - C_{20}^{7} = \frac{20!}{6! \cdot 14!} = \frac{15 \cdot 16 \cdot 17 \cdot 18 \cdot 19 \cdot 20}{720} = 38\ 760\);
8) \(C_{25}^{9} - C_{24}^{9} = C_{24}^{8} + C_{24}^{9} - C_{24}^{9} =\)
\(= \frac{24!}{8! \cdot 16!} = \frac{17 \cdot 18 \cdot 19 \cdot 20 \cdot 21 \cdot 22 \cdot 23 \cdot 24}{8!} = 735\ 471\);
9) \(C_{15}^{10} - C_{14}^{9} = C_{14}^{9} + C_{14}^{10} - C_{14}^{9} = \frac{14!}{10! \cdot 4!} = \frac{11 \cdot 12 \cdot 13 \cdot 14}{4!} = 1\ 001\);
10) \(C_{13}^{8} - C_{12}^{7} = C_{12}^{8} + C_{12}^{7} - C_{12}^{7} = \frac{12!}{8! \cdot 4!} = \frac{9 \cdot 10 \cdot 11 \cdot 12}{4!} = 495\).
