UPSC NDA II 2015 Mathematics Paper

Concept:The sum of all binomial coefficients in the expansion of $(1+x)^m$ is $2^m$, obtained by substituting $x=1$.Explanation:The question asks for the sum of the coefficients of all terms in the expansion of $(1+x)^{2n+1}$.To find this sum, set $x=1$ in the binomial expansion.Then $(1+1)^{2n+1} = 2^{2n+1}$ gives the required sum.Simplify $2^{2n+1}$ as $2^{2n} \times 2 = (2^2)^n \times 2 = 4^n \times 2 = 2 \times 4^n$.Thus the sum of the coefficients equals $2 \times 4^n$.This matches option C.Answer:Option C: $2 \times 4^n$