Binomial Theorem
Properties of Binomial Coefficients
Premium Question
Grade 11
Question:
For $n \geq 1$, the value of $\binom{n}{0} - \binom{n}{1} + \binom{n}{2} - \cdots + (-1)^n \binom{n}{n}$ is:
(1) $0$
(2) $1$
(3) $2^n$
(4) $2^{n-1}$
Step-by-Step Solution
Key Concept: Substitute $x = -1$ into the expansion $(1 + x)^n = \sum \binom{n}{r} x^r$ and simplify the resulting power of $(1 - 1)$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (1)