Binomial Theorem
Sum of Alternating Binomial Coefficients
nta_pyq_2023_apr
Grade 11
Question:
The sum of the coefficients of the first 50 terms in the binomial expansion of $(1-x)^{100}$, is equal to
${}^{101}C_{50}$
${}^{99}C_{49}$
$-{}^{101}C_{50}$
$-{}^{99}C_{49}$
Step-by-Step Solution
Key Concept: First 50 terms have coefficients $S=\binom{100}{0}-\binom{100}{1}+\cdots-\binom{100}{49}$. Use $(1-x)^{100}|_{x=1}=0$ and symmetry of binomial coefficients.
$2S+\binom{100}{50}=0\Rightarrow S=-\frac{\binom{100}{50}}{2}=-\binom{99}{49}$.
Correct Answer: 4