Let \[{\rm{S}}\] be a set containing \[{\rm{n}}\] elements and we select 2 subsets \[{\rm{A}}\] and \[{\rm{B}}\] of \[{\rm{S}}\] at random then the probability that \[A \cup B = S\] and \[A \cap B = \varphi \] is
A. \[{2^n}\]
B. \[{n^2}\]
C. \[1/{\rm{n}}\]
D. \[1/{2^n}\]

Hint: In this question, we must calculate the chance that \[A \cup B = S\] and \[A \cap B = \varphi \] where \[{\rm{A}}\] and \[{\rm{B}}\] have the same number of items, such that A and B are subsets of \[{\rm{S}}\] chosen at random from a set of n elements. For n items, we shall apply the concept of total number of elements for a certain subset. We must apply the concept of probability in order to get the desired answer.



Formula Used:The set’s number of subset \[X\] is
\[{2^n}\]



Complete step by step solution:We have been given an information in the question that,
Let \[{\rm{S}}\] be a set containing \[{\rm{n}}\] elements and we select 2 subsets \[{\rm{A}}\] and \[{\rm{B}}\] of \[{\rm{S}}\] at random.
And we are to calculate the probability that
\[A \cup B = S\] And \[A \cap B = \varphi \]
Given that \[X\] is a set with n items, just let the elements of \[X\] be
\[X = \{ {a_1},{a_2},{a_3},.......{a_n}\} \]
We are provided that \[S\] has \[n\] elements and two sets \[A\] and \[B\] are taken.
We have been already known that,
Now that we understand that the number of subsets of a set with n items equals
\[{2^n}\]
So, the set \[X\] ‘s subset can be determined as
\[{2^n}\]
The number of ways to choose two sets so that their union is \[S\] and their intersection is nullset is one.
Therefore, the required probability is
\[\dfrac{1}{{2n}}\]



Option ‘A’ is correct



Note: Students should remember that the number of subsets is \[{2^n}\] of an n-element set. Elements that comprise a set are a Subset. It is critical that the subset exclusively contain (include) entries from the set from which it is defined. So, one should be very cautious in applying probability concepts in order to get the desired answer. Applying wrong concepts will lead to wrong solution.