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 the elements of all possible subsets \( B \) of size 11 that can be formed from the set \( A = \{ 1, 2, 3, \ldots, 22 \} \).

### Step-by-Step Solution:

1. **Identify the Set A**:
The set \( A \) consists of the integers from 1 to 22.
\[ A = \{ 1, 2, 3, \ldots, 22 \} \]

2. **Calculate the Sum of Elements in Set A**:
We can use the formula for the sum of the first \( n \) natural numbers:
\[ \text{Sum} = \frac{n(n + 1)}{2} \]
Here, \( n = 22 \):
\[ \text{Sum of A} = \frac{22 \times 23}{2} = 253 \]

3. **Determine the Number of Ways to Choose Subset B**:
The number of ways to choose a subset \( B \) of size 11 from set \( A \) is given by the binomial coefficient:
\[ \binom{22}{11} \]

4. **Calculate the Contribution of Each Element in A**:
Each element in \( A \) will appear in half of the subsets of size 11. Since we are choosing 11 elements from 22, each element will be included in:
\[ \frac{1}{2} \times \binom{22}{11} \]
subsets of size 11.

5. **Sum of Elements of All Possible Subsets B**:
The total contribution of all elements in \( A \) to the sum of all subsets \( B \) can be calculated as:
\[ \text{Total Sum} = \binom{22}{11} \times \frac{253}{2} \]

6. **Final Calculation**:
We can express this as:
\[ \text{Total Sum} = \frac{253}{2} \times \binom{22}{11} \]

### Conclusion:
The sum of the elements of all possible subsets \( B \) is:
\[ \frac{253}{2} \times \binom{22}{11} \]