Bài tập nâng cao · Bài 26
Bài tập nâng cao Nhị thức Newton · Bài 26
Tổng $C_{2019}^0+C_{2019}^1+C_{2019}^2+C_{2019}^3+\ldots+C_{2019}^{2016}+C_{2019}^{2017}$ có giá trị bằng:
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Lời giải
Ta có
$C_{2019}^0+C_{2019}^1+C_{2019}^2+C_{2019}^3+\ldots+C_{2019}^{2016}+C_{2019}^{2017}+C_{2019}^{2018}+C_{2019}^{2019}=2^{2019}$.
$\Leftrightarrow C_{2019}^0+C_{2019}^1+C_{2019}^2+C_{2019}^3+\ldots+C_{2019}^{2016}+C_{2019}^{2017}=2^{2019}-C_{2019}^{2018}-C_{2019}^{2019}$
$\Leftrightarrow C_{2019}^0+C_{2019}^1+C_{2019}^2+C_{2019}^3+\ldots+C_{2019}^{2016}+C_{2019}^{2017}=2^{2019}-2019-1$
$\Leftrightarrow C_{2019}^0+C_{2019}^1+C_{2019}^2+C_{2019}^3+\ldots+C_{2019}^{2016}+C_{2019}^{2017}=2^{2019}-2020$.
$C_{2019}^0+C_{2019}^1+C_{2019}^2+C_{2019}^3+\ldots+C_{2019}^{2016}+C_{2019}^{2017}+C_{2019}^{2018}+C_{2019}^{2019}=2^{2019}$.
$\Leftrightarrow C_{2019}^0+C_{2019}^1+C_{2019}^2+C_{2019}^3+\ldots+C_{2019}^{2016}+C_{2019}^{2017}=2^{2019}-C_{2019}^{2018}-C_{2019}^{2019}$
$\Leftrightarrow C_{2019}^0+C_{2019}^1+C_{2019}^2+C_{2019}^3+\ldots+C_{2019}^{2016}+C_{2019}^{2017}=2^{2019}-2019-1$
$\Leftrightarrow C_{2019}^0+C_{2019}^1+C_{2019}^2+C_{2019}^3+\ldots+C_{2019}^{2016}+C_{2019}^{2017}=2^{2019}-2020$.