A) \[{{2}^{n-1}}\]
B) \[{{2}^{n}}\]
C) \[{{2}^{n+1}}\]
D) \[{{2}^{2n}}\]
Correct Answer: D
Solution :
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. \ Required number of subsets \[={{\,}^{2n+1}}{{C}_{n+1}}+{{\,}^{2n+1}}{{C}_{n+2}}+....+{{\,}^{2n+1}}{{C}_{2n}}+{{\,}^{2n+1}}{{C}_{2n+1}}\] \[={{\,}^{2n+1}}{{C}_{n}}+{{\,}^{2n+1}}{{C}_{n-1}}+...+{{\,}^{2n+1}}{{C}_{1}}+{{\,}^{2n+1}}{{C}_{0}}\] \[={{\,}^{2n+1}}{{C}_{0}}+{{\,}^{2n+1}}{{C}_{1}}+{{\,}^{2n+1}}{{C}_{2}}+...+{{\,}^{2n+1}}{{C}_{n-1}}+{{\,}^{2n+1}}{{C}_{n}}\] \[=\frac{1}{2}\left[ {{(1+1)}^{2n+1}} \right]\]\[=\frac{1}{2}[{{2}^{2n+1}}]={{2}^{2n}}\].
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