Binomial Theorem
Properties of Binomial Coefficients
Premium Question
Grade 11
Question:
The value of $\binom{n}{0} + \binom{n}{2} + \binom{n}{4} + \cdots$ (sum of binomial coefficients of even rank) equals:
(1) $2^{n-1}$
(2) $2^n$
(3) $2^{n-2}$
(4) $n \cdot 2^{n-2}$
Step-by-Step Solution
Key Concept: Add the expansions of $(1 + x)^n$ and $(1 - x)^n$ at $x = 1$ — the odd-rank terms cancel, leaving exactly twice the sum of the even-rank coefficients.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (1)