The sum, of the coefficients of the first 50 terms in the binomial expansion of $(1-x)^{100}$, is equal to
Solution
$$
\begin{aligned}
& \left({ }^{100} C_0-{ }^{100} C_1+{ }^{100} C_2-\ldots . .{ }^{100} C_{49}\right)+{ }^{100} C_{50} \\\\
& +\left(-{ }^{100} C_{51}+{ }^{100} C_{52}-\ldots .+{ }^{100} C_{100}\right)=0 \\\\
& \lambda_1+{ }^{100} C_{50}+\lambda_2=0 \\\\
& \lambda_1=-\frac{1}{2}{ }^{100} C_{50} \quad\left(\because \lambda_1=\lambda_2\right) \\\\
& =-{ }^{99} C_{49}
\end{aligned}
$$
About this question
Subject: Mathematics · Chapter: Binomial Theorem · Topic: Binomial Expansion
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