A set contains 2n+1 elements. The number of subsets of this set containing more than n elements :

Options

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

Answer

To solve the problem of finding the number of subsets of a set containing \(2n + 1\) elements that contain more than \(n\) elements, we can follow these steps:

### Step 1: Calculate the total number of subsets
The total number of subsets of a set with \(m\) elements is given by \(2^m\). In this case, the set has \(2n + 1\) elements. Therefore, the total number of subsets is:
\[ 2^{2n + 1} \]

### Step 2: Calculate the number of subsets with \(n\) or fewer elements
To find the number of subsets that contain more than \(n\) elements, we first need to calculate the number of subsets that contain \(n\) or fewer elements. This includes subsets with \(0\) to \(n\) elements.

The number of subsets with \(k\) elements from a set of \(m\) elements is given by the binomial coefficient \(\binom{m}{k}\). Thus, the number of subsets with \(0\) to \(n\) elements is:
\[ \sum_{k=0}^{n} \binom{2n + 1}{k} \]

### Step 3: Use the property of binomial coefficients
By the symmetry property of binomial coefficients, we know that:
\[ \sum_{k=0}^{m} \binom{m}{k} = 2^m \]
For our case, \(m = 2n + 1\), so:
\[ \sum_{k=0}^{2n + 1} \binom{2n + 1}{k} = 2^{2n + 1} \]
This can be split into two parts:
\[ \sum_{k=0}^{n} \binom{2n + 1}{k} + \sum_{k=n+1}^{2n + 1} \binom{2n + 1}{k} = 2^{2n + 1} \]
Since the set is symmetric, we have:
\[ \sum_{k=0}^{n} \binom{2n + 1}{k} = \sum_{k=n+1}^{2n + 1} \binom{2n + 1}{k} \]
Let \(S\) be the sum of subsets with \(0\) to \(n\) elements:
\[ S = \sum_{k=0}^{n} \binom{2n + 1}{k} \]
Thus, we can write:
\[ S + S = 2^{2n + 1} \implies 2S = 2^{2n + 1} \implies S = 2^{2n} \]

### Step 4: Calculate the number of subsets with more than \(n\) elements
Now, to find the number of subsets with more than \(n\) elements, we subtract the number of subsets with \(n\) or fewer elements from the total number of subsets:
\[ \text{Number of subsets with more than } n \text{ elements} = 2^{2n + 1} - 2^{2n} \]

### Step 5: Simplify the expression
We can factor out \(2^{2n}\) from the expression:
\[ 2^{2n + 1} - 2^{2n} = 2^{2n}(2 - 1) = 2^{2n} \]

### Final Answer
Thus, the number of subsets of the set containing more than \(n\) elements is:
\[ \boxed{2^{2n}} \]