Two subsets A and B of a set containing n elements are chosen at random. The probability that `A sube B` is

Options

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

Answer

To solve the problem of finding the probability that subset A is a subset of subset B when both are chosen from a set containing n elements, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Total Number of Subsets**:
Each element in a set can either be included in a subset or not. Therefore, for a set with n elements, the total number of subsets is \(2^n\). This is because each of the n elements has 2 choices: to be included or not.

2. **Calculating the Total Ways to Choose Subsets A and B**:
Since both subsets A and B can be chosen independently, the total number of ways to choose both subsets A and B is:
\[ 2^n \times 2^n = 2^{2n} = 4^n \]

3. **Understanding the Condition A ⊆ B**:
The condition \(A \subseteq B\) means that every element in subset A must also be in subset B. For each element in the original set, we have the following choices:
- The element can be in neither A nor B.
- The element can be in B but not in A.
- The element can be in both A and B.

Thus, for each element, we have 3 choices:
- Not included in A or B (0)
- Included in B but not in A (1)
- Included in both A and B (2)

4. **Calculating the Favorable Cases**:
Since there are n elements and each has 3 choices, the total number of favorable cases where \(A \subseteq B\) is:
\[ 3^n \]

5. **Calculating the Probability**:
The probability that \(A \subseteq B\) is given by the ratio of the number of favorable cases to the total number of cases:
\[ P(A \subseteq B) = \frac{\text{Number of favorable cases}}{\text{Total number of cases}} = \frac{3^n}{4^n} = \left(\frac{3}{4}\right)^n \]

### Final Answer:
The probability that subset A is a subset of subset B is:
\[ \left(\frac{3}{4}\right)^n \]