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If two subsets A and B are selected at random from a set S containing n elements, then the probability that $A \cap B = ϕ and A \cup B = S, is$

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$\frac{1}{2^{n}}$

$\frac{1}{2^{n + 1}}$

$2^{n}$

$\frac{1}{2^{n} \times 2^{n}}$

answer is A.

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Detailed Solution

$A = [n_{c_{0}} + n_{c_{1}} + n_{c_{2}} + \cdot \cdot \cdot \cdot + n_{c_{n}}]$

$B = 2^{n}$

$A \cap B = ϕ and A \cup B = S$

Probability $= \frac{2^{n}}{2^{n} . 2^{n}}$

$= \frac{1}{2^{n}}$

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