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  • The number of subsets of the set <img alt="\mathrm{\operatorname{set} A=\left\{a_{1}, a_{2}\right., \left.\ldots, a_{n}\right\}}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7B%5

The number of subsets of the set \mathrm{\operatorname{set} A=\left\{a_{1}, a_{2}\right., \left.\ldots, a_{n}\right\}} which contain even number of elements is

Option: 1

\mathrm{2^{n-1}}


Option: 2

\mathrm{2^{n}-1}


Option: 3

\mathrm{2^{n-2}}


Option: 4

\mathrm{2^{n}}


Answers (1)

best_answer

 For eav' of the first \mathrm{(n-1)} elements \mathrm{a_{1}, a_{2}, \ldots, a_{n-1}} we have two choices: either \mathrm{a_{i}(1 \leq i \leq n-1)}  lies in the subset or \mathrm{a_{i}} doesn't lie in the subset. For the last element we have just one choice. If even number of elements have already been selected, we do not include \mathrm{a_{n}} in the subset, otherwise (when odd number of elements have been selected), we include it in the subset.

Thus, the number of subsets of \mathrm{A=\left\{a_{1}, a_{2}, \ldots, a_{n}\right\}}which contain even number of elements is equal to \mathrm{2^{n-1}}.

Posted by

Ritika Kankaria

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