The number of subsets of a set containing n elements is :

Options

A. n
B. `2^(n)-1`
C. `2^(n-1)`
D. `2^(n)`

Answer

To find the number of subsets of a set containing \( n \) elements, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
We need to determine how many subsets can be formed from a set that has \( n \) elements.

2. **Choosing Elements**:
For a set with \( n \) elements, we can choose:
- 0 elements (which gives us the empty set),
- 1 element,
- 2 elements,
- and so on, up to \( n \) elements (which gives us the set itself).

3. **Using Combinations**:
The number of ways to choose \( k \) elements from \( n \) elements is given by the binomial coefficient \( \binom{n}{k} \). Therefore, the number of subsets can be expressed as:
\[ \text{Number of subsets} = \binom{n}{0} + \binom{n}{1} + \binom{n}{2} + \ldots + \binom{n}{n} \]

4. **Applying the Binomial Theorem**:
According to the binomial theorem, we know that:
\[ (1 + x)^n = \sum_{k=0}^{n} \binom{n}{k} x^k \]
If we set \( x = 1 \), we get:
\[ (1 + 1)^n = \sum_{k=0}^{n} \binom{n}{k} \cdot 1^k = \sum_{k=0}^{n} \binom{n}{k} \]
This simplifies to:
\[ 2^n = \sum_{k=0}^{n} \binom{n}{k} \]

5. **Conclusion**:
Therefore, the total number of subsets of a set containing \( n \) elements is:
\[ \text{Number of subsets} = 2^n \]

### Final Answer:
The number of subsets of a set containing \( n \) elements is \( 2^n \).

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