b, 15c8 + 15c9 + 15c10 + ... + 15c15
Xét khai triển:
\(\left(1+x\right)^{2021}=C_{2021}^0+C_{2021}^1x+...+C_{2021}^{2020}x^{2020}+C_{2021}^{2021}x^{2021}\)
Thay \(x=1\)
\(\Rightarrow2^{2021}=C_{2021}^0+C_{2021}^1+...+C_{2021}^{2021}\) (1)
Thay \(x=-1\)
\(\Rightarrow0=C_{2021}^0-C_{2021}^1+C_{2021}^2-C_{2021}^3+...+C_{2021}^{2020}-C_{2021}^{2021}\)
\(\Rightarrow C_{2021}^0+C_{2021}^2+...+C_{2021}^{2020}=C_{2021}^1+...+C_{2021}^{2021}\) (2)
(1); (2) \(\Rightarrow2\left(C_{2021}^0+C_{2021}^2+...+C_{2021}^{2020}\right)=2^{2021}\)
\(\Rightarrow T=C_{2021}^0+C_{2021}^2+...+C_{2021}^{2020}=2^{2020}\)
b.
Xét khai triển: \(\left(1+x\right)^{15}=C_{15}^0+C_{15}^1x+...+C_{15}^{15}x^{15}\)
Cho \(x=1\Rightarrow C_{15}^0+C_{15}^1+...+C_{15}^{15}=2^{15}\)
Mặt khác: \(\left\{{}\begin{matrix}C_{15}^0=C_{15}^{15}\\C_{15}^1=C_{15}^{14}\\...\\C_{15}^7=C_{15}^8\end{matrix}\right.\) \(\Rightarrow2\left(C_{15}^8+C_{15}^9+...+C_{15}^{15}\right)=2^{15}\)
\(\Rightarrow S=2^{14}\)
