let S = {2,4,6.............20), what is the number of subsets does S have?

Options

A. 10
B. 20
C. 512
D. 1024

Answer

To solve the problem of finding the number of subsets of the set \( S = \{2, 4, 6, \ldots, 20\} \), we can follow these steps:

### Step 1: Identify the elements of the set \( S \)
The set \( S \) consists of even numbers from 2 to 20. We can list them:
\[ S = \{2, 4, 6, 8, 10, 12, 14, 16, 18, 20\} \]

### Step 2: Count the number of elements in the set \( S \)
We can see that the elements are:
- 2
- 4
- 6
- 8
- 10
- 12
- 14
- 16
- 18
- 20

Thus, the total number of elements \( n \) in the set \( S \) is:
\[ n = 10 \]

### Step 3: Use the formula for the number of subsets
The formula for the number of subsets of a set with \( n \) elements is given by:
\[ \text{Number of subsets} = 2^n \]
Substituting the value of \( n \):
\[ \text{Number of subsets} = 2^{10} \]

### Step 4: Calculate \( 2^{10} \)
Now we calculate \( 2^{10} \):
\[ 2^{10} = 1024 \]

### Step 5: Conclusion
Therefore, the number of subsets of the set \( S \) is:
\[ \text{Number of subsets} = 1024 \]

### Final Answer
The number of subsets of the set \( S \) is \( 1024 \).

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