Let set ` A = { 1, 2, 3, ….., 22} ` . Set B is a subset of A and B has exactly 11 elements, find the sum of elements of all possible subsets B .

Options

A. ` 252^(21)C_(11) `
B. ` 230^(21)C_(10) `
C. ` 253^(21)C_(9) `
D. ` 253^(21)C_(10) `

Answer

To solve the problem, we need to find the sum of elements of all possible subsets B of set A, where set B has exactly 11 elements.

### Step-by-Step Solution:

1. **Identify Set A**:
Set A consists of the first 22 natural numbers:
\[ A = \{1, 2, 3, \ldots, 22\} \]

2. **Calculate the Total Sum of Set A**:
The sum of the first n natural numbers can be calculated using the formula:
\[ S = \frac{n(n + 1)}{2} \]
For our case, \( n = 22 \):
\[ S = \frac{22 \times 23}{2} = 253 \]

3. **Determine the Number of Ways to Choose Subset B**:
We need to choose 11 elements from the 22 elements in set A. The number of ways to choose 11 elements from 22 is given by the combination formula:
\[ \binom{n}{r} = \frac{n!}{r!(n - r)!} \]
Here, \( n = 22 \) and \( r = 11 \):
\[ \binom{22}{11} = \frac{22!}{11! \cdot 11!} \]

4. **Calculate the Contribution of Each Element**:
Each element in set A will appear in several subsets B. Specifically, each element will be included in half of the subsets of size 11. The number of subsets of size 11 that include a specific element \( k \) can be calculated by choosing the remaining 10 elements from the other 21 elements:
\[ \binom{21}{10} \]

5. **Sum of All Elements in All Subsets B**:
The total contribution of all elements in set A to all subsets B can be calculated by multiplying the sum of all elements in A by the number of subsets B that each element appears in:
\[ \text{Total Sum} = \text{Sum of A} \times \text{Number of subsets B with each element} \]
Thus, the total sum is:
\[ \text{Total Sum} = 253 \times \binom{21}{10} \]

6. **Calculate \(\binom{21}{10}\)**:
Using the combination formula:
\[ \binom{21}{10} = \frac{21!}{10! \cdot 11!} = 352716 \]

7. **Final Calculation**:
Now, substituting back into the total sum:
\[ \text{Total Sum} = 253 \times 352716 = 89210028 \]

### Final Answer:
The sum of elements of all possible subsets B is:
\[ \boxed{89210028} \]