Which of the following cannot be the number of elements in the power set of any finite set ?

Options

A. 26
B. 32
C. 6
D. 16

Answer

To determine which of the given numbers cannot be the number of elements in the power set of any finite set, we need to use the formula for the number of elements in a power set. The number of elements in the power set of a set with \( n \) elements is given by:

\[ \text{Number of elements in power set} = 2^n \]

where \( n \) is a non-negative integer. This means that the number of elements in a power set must be a power of 2.

Now, let's analyze each of the given options: 26, 32, 6, and 12.

1. **Check 26**:
- We need to see if 26 can be expressed as \( 2^n \) for some non-negative integer \( n \).
- The powers of 2 are: \( 1, 2, 4, 8, 16, 32, \ldots \)
- 26 is not in this list, so it cannot be expressed as \( 2^n \).

2. **Check 32**:
- 32 can be expressed as \( 2^5 \) (since \( 2^5 = 32 \)).
- Therefore, 32 can be the number of elements in a power set.

3. **Check 6**:
- 6 cannot be expressed as \( 2^n \) for any non-negative integer \( n \).
- The closest powers of 2 are \( 4 (2^2) \) and \( 8 (2^3) \), and 6 is not a power of 2.

4. **Check 12**:
- 12 cannot be expressed as \( 2^n \) for any non-negative integer \( n \).
- The closest powers of 2 are \( 8 (2^3) \) and \( 16 (2^4) \), and 12 is not a power of 2.

From this analysis, we conclude that the numbers that cannot be the number of elements in the power set are **26, 6, and 12**.

### Final Answer:
The numbers that cannot be the number of elements in the power set of any finite set are **26, 6, and 12**.