If `A_1 sub A_2 sub A_3 sub ...... sub A_50` and `n(A_x) = x -1`, then find `n[nnn_(x=11)^50 A_x]`

Options

A. 99
B. 98
C. 100
D. 10

Answer

To solve the problem, we need to find the number of elements in the intersection of the sets \( A_x \) for \( x \) ranging from 11 to 50, given that \( A_1 \subset A_2 \subset A_3 \subset \ldots \subset A_{50} \) and \( n(A_x) = x - 1 \).

### Step-by-Step Solution:

1. **Understand the Given Information**:
- We know that \( A_1 \subset A_2 \subset A_3 \subset \ldots \subset A_{50} \).
- The number of elements in each set \( A_x \) is given by the formula \( n(A_x) = x - 1 \).

2. **Determine the Number of Elements in Each Set**:
- For \( A_{11} \):
\[ n(A_{11}) = 11 - 1 = 10 \]
- For \( A_{12} \):
\[ n(A_{12}) = 12 - 1 = 11 \]
- For \( A_{13} \):
\[ n(A_{13}) = 13 - 1 = 12 \]
- Continuing this way, we find that for \( A_{50} \):
\[ n(A_{50}) = 50 - 1 = 49 \]

3. **Identify the Intersection**:
- Since \( A_1 \subset A_2 \subset \ldots \subset A_{50} \), the intersection of all sets \( A_{11}, A_{12}, \ldots, A_{50} \) will contain only the elements that are present in the smallest set, which is \( A_{11} \).
- Therefore, the intersection \( \bigcap_{x=11}^{50} A_x = A_{11} \).

4. **Find the Number of Elements in the Intersection**:
- Since the intersection is equal to \( A_{11} \), we need to find the number of elements in \( A_{11} \):
\[ n\left(\bigcap_{x=11}^{50} A_x\right) = n(A_{11}) = 10 \]

### Final Answer:
\[ n\left(\bigcap_{x=11}^{50} A_x\right) = 10 \]