Question

A set contains 2n+1 elements. The number of subsets of this set containing more than n elements is equal to

• A. $2^{n-1}$
• B. $2^n$
• C. $2^{n+1}$
• D. $2^{2n}$

Answer 1

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

Let the original set contains (2n+1) elements, then subsets of this set containing more than n elements, i.e., subsets containing (n+1) elements, (n+2) elements,....(2n+1) elements.
$\therefore$ Required number of subsets
${=}^{2n+1}{C}_{n+1}{+}^{2n+1}{C}_{n+2}+...{+}^{2n+1}{C}_{2n}{+}^{2n+1}{C}_{2n+1}$
${=}^{2n+1}{C}_n{+}^{2n+1}{C}_{n-1}+...{+}^{2n+1}{C}_1{+}^{2n+1}{C}_0$
${=}^{2n+1}{C}_0{+}^{2n+1}{C}_1{+}^{2n+1}{C}_2+...{+}^{2n+1}{C}_{n-1}{+}^{2n+1}{C}_n$
$=\frac{1}{2}\left[\right(1+1{)}^{2n+1}]=\frac{1}{2}[2^{2n+1}]=2^{2n}$.