Number of subsets of a $(2n+1)$-element set containing at least half the elements? A set contains $(2n+1)$ elements.The number of subsets of this set containing more than $n$ elements is equal to $2^{2n}$. 
How is this possible, if the number of subsets of a set is $2^{n}$.

Consider a subset of a set of $2n+1$ elements and its complement. Exactly one of the two has more than $n$ elements. Since the total number of subsets is $2^{2n+1}$ and they come in complementary pairs, exactly half, i.e. $2^{2n}$, have more than $n$ elements.
