The number of all subsets of a set containing `2n+1` elements which contains more than `n` elements is

Options

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

Explanation

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

### Step 1: Understand the Total Number of Subsets
A set with \(m\) elements has a total of \(2^m\) subsets. Therefore, for a set with \(2n + 1\) elements, the total number of subsets is:
\[ 2^{2n + 1} \]

**Hint:** Remember that each element can either be included or excluded from a subset, leading to \(2^m\) combinations.

### Step 2: Calculate the Number of Subsets with \(n\) or Fewer Elements
We need to find the number of subsets that contain \(n\) or fewer elements. This can be calculated using the binomial coefficient:
\[ \sum_{k=0}^{n} \binom{2n + 1}{k} \]
This sum represents the total number of subsets with \(0\) to \(n\) elements.

**Hint:** The binomial coefficient \(\binom{m}{k}\) gives the number of ways to choose \(k\) elements from \(m\) elements.

### Step 3: Use the Symmetry of Binomial Coefficients
By the symmetry property of binomial coefficients, we have:
\[ \binom{2n + 1}{k} = \binom{2n + 1}{2n + 1 - k} \]
This means that the number of subsets with more than \(n\) elements is equal to the number of subsets with fewer than \(n\) elements.

**Hint:** This property helps in simplifying the calculations by relating subsets of different sizes.

### Step 4: Calculate the Number of Subsets with More Than \(n\) Elements
The number of subsets with more than \(n\) elements can be found by subtracting the number of subsets with \(n\) or fewer elements from the total number of subsets:
\[ \text{Number of subsets with more than } n \text{ elements} = 2^{2n + 1} - \sum_{k=0}^{n} \binom{2n + 1}{k} \]

Using the symmetry property:
\[ \sum_{k=0}^{n} \binom{2n + 1}{k} = \sum_{k=n+1}^{2n + 1} \binom{2n + 1}{k} \]
Thus, we can express the total number of subsets as:
\[ 2^{2n + 1} = 2 \sum_{k=0}^{n} \binom{2n + 1}{k} \]
This leads us to:
\[ \sum_{k=0}^{n} \binom{2n + 1}{k} = \frac{2^{2n + 1}}{2} = 2^{2n} \]

### Step 5: Final Calculation
Now substituting back, we find:
\[ \text{Number of subsets with more than } n \text{ elements} = 2^{2n + 1} - 2^{2n} = 2^{2n} (2 - 1) = 2^{2n} \]

Thus, the number of all subsets of a set containing \(2n + 1\) elements which contain more than \(n\) elements is:
\[ \boxed{2^{2n}} \]