Решебник по алгебре и начала математического анализа 11 класс Колягин Задание 455

\[1)\ C_{15}^{12} + C_{15}^{13} = C_{15}^{12} + C_{15}^{12 + 1} =\]

\[= C_{15 + 1}^{12 + 1} = C_{16}^{13} = \frac{16!}{13!(16 - 13)!} =\]

\[= \frac{16 \bullet 15 \bullet 14 \bullet 13!}{13! \bullet 3!} = \frac{16 \bullet 15 \bullet 14}{3 \bullet 2} =\]

\[= 560.\]

\[2)\ C_{11}^{3} + C_{11}^{2} = C_{11}^{2} + C_{11}^{2 + 1} =\]

\[= C_{11 + 1}^{2 + 1} = C_{12}^{3} = \frac{12!}{3!(12 - 3)!} =\]

\[= \frac{12 \bullet 11 \bullet 10 \bullet 9!}{3 \bullet 2 \bullet 9!} = \frac{12 \bullet 11 \bullet 10}{3 \bullet 2} =\]

\[= 220.\]

\[3)\ C_{21}^{4} - C_{20}^{4} = C_{20 + 1}^{3 + 1} - C_{20}^{4} =\]

\[= C_{20}^{3} + C_{20}^{4} - C_{20}^{4} = C_{20}^{3} =\]

\[= \frac{20!}{3!(20 - 3)!} = \frac{20 \bullet 19 \bullet 18 \bullet 17!}{3 \bullet 2 \bullet 17!} =\]

\[= \frac{20 \bullet 19 \bullet 18}{3 \bullet 2} = 1140.\]

\[4)\ C_{101}^{3} - C_{100}^{3} = C_{100 + 1}^{2 + 1} - C_{100}^{3} =\]

\[= C_{100}^{2} + C_{100}^{3} - C_{100}^{3} = C_{100}^{2} =\]

\[{= \frac{100!}{2!(100 - 2)!} = \frac{100 \bullet 99 \bullet 98!}{2 \bullet 98!} = }{= \frac{100 \bullet 99}{2} = 4950.}\]