Question

A set contains $(2n+1)$ elements. Then the number of subsets of the set which contains at most $n$ elements is

• A. $2^n$
• B. $2^{2n}$
• C. $2^{3n}$
• D. $2^{4n}$

Answer 1

The correct option is B $2^{2n}$

The number of subsets of a set which contains at most $n$ elements is $N={}^{2n+1}{C}_0+{}^{2n+1}{C}_1+{}^{2n+1}{C}_2+...+{}^{2n+1}{C}_n\Rightarrow N=2^{2n+1}-N\Rightarrow 2N=2^{2n+1}\Rightarrow N=\frac{1}{2}\cdot 2^{2n+1}\Rightarrow N=2^{2n}$