If two subsets A and B of set S containing n elements are selected at random, then the probabit `AnnB=phi and AuuB=S` is

Options

A. `(1)/(2)`
B. `(1)/(2^n)`
C. `((3)/(4))^4`
D. `(1)/(3^n)`.

Answer

To solve the problem, we need to find the probability that two randomly selected subsets \( A \) and \( B \) of a set \( S \) with \( n \) elements satisfy the conditions \( A \cap B = \emptyset \) (the intersection of \( A \) and \( B \) is empty) and \( A \cup B = S \) (the union of \( A \) and \( B \) is the entire set \( S \)).

### Step-by-step Solution:

1. **Understanding the Set \( S \)**:
- Let \( S \) be a set containing \( n \) elements. The total number of subsets of \( S \) is \( 2^n \).

2. **Conditions for Subsets \( A \) and \( B \)**:
- We need \( A \cap B = \emptyset \), which means that \( A \) and \( B \) do not share any elements.
- We also need \( A \cup B = S \), which means every element of \( S \) must be included in either \( A \) or \( B \).

3. **Choosing Elements for \( A \) and \( B \)**:
- For each element in \( S \), there are three choices:
- The element can be in subset \( A \).
- The element can be in subset \( B \).
- The element can be in neither subset (but this option is not allowed since \( A \cup B \) must equal \( S \)).
- Therefore, each element must be assigned to either \( A \) or \( B \), leading to two choices per element.

4. **Total Combinations**:
- Since each of the \( n \) elements can independently be assigned to either \( A \) or \( B \), the total number of valid combinations for subsets \( A \) and \( B \) is \( 2^n \).

5. **Total Outcomes**:
- The total number of ways to select any two subsets \( A \) and \( B \) from \( S \) is \( 2^n \times 2^n = 2^{2n} \).

6. **Calculating the Probability**:
- The probability \( P \) that two randomly selected subsets \( A \) and \( B \) satisfy both conditions is given by:
\[ P = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{2^n}{2^{2n}} = \frac{1}{2^n} \]

### Final Answer:
The probability that \( A \cap B = \emptyset \) and \( A \cup B = S \) is \( \frac{1}{2^n} \).