if A is a finite set having n elements , then P(A) has

Options

A. 2n elements
B. `2^(n)` elements
C. n elements
D. none of these

Answer

To solve the problem, we need to determine the number of elements in the power set \( P(A) \) of a finite set \( A \) that has \( n \) elements.

### Step-by-Step Solution:

1. **Understanding the Set \( A \)**:
- Let \( A \) be a finite set with \( n \) elements. This means that the total number of elements in set \( A \) is \( n \).

**Hint**: Remember that a finite set has a specific count of elements.

2. **Definition of Power Set**:
- The power set \( P(A) \) is defined as the set of all possible subsets of \( A \). This includes the empty set and \( A \) itself.

**Hint**: Recall that the power set includes every possible combination of the elements in the original set.

3. **Formula for the Number of Subsets**:
- The number of subsets of a set with \( n \) elements is given by the formula \( 2^n \). This is because each element can either be included in a subset or not, leading to \( 2 \) choices (include or exclude) for each of the \( n \) elements.

**Hint**: Think about the binary choices (in or out) for each element when forming subsets.

4. **Conclusion about the Power Set**:
- Since the power set \( P(A) \) contains all subsets of \( A \), the number of elements (subsets) in the power set \( P(A) \) is \( 2^n \).

**Hint**: Remember that the total number of subsets is equal to the number of elements in the power set.

### Final Answer:
The power set \( P(A) \) has \( 2^n \) elements.