Binomial Theorem
Properties of Binomial Coefficients
JEE Main 2021
Grade 11
Question:
If $(1 + x)^n = \sum_{r=0}^n \binom{n}{r} x^r$, then $\sum_{r=0}^n \binom{n}{r}$ equals:
(1) $2^n$
(2) $2^{n-1}$
(3) $n \cdot 2^{n-1}$
(4) $n^2$
Step-by-Step Solution
Key Concept: Substitute a particularly convenient value of $x$ into the binomial expansion $(1 + x)^n = \sum \binom{n}{r} x^r$ so that every term on the right reduces to just its coefficient $\binom{n}{r}$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (1)