To solve the problem, we need to find the number of elements in the range of \( n(A \Delta B) \), where \( A \Delta B \) is the symmetric difference of sets \( A \) and \( B \).
### Step-by-step Solution:
1. **Understanding the Symmetric Difference**:
The symmetric difference \( A \Delta B \) is defined as:
\[
A \Delta B = (A - B) \cup (B - A)
\]
This means it includes all elements that are in either \( A \) or \( B \) but not in both.
2. **Using the Formula for Symmetric Difference**:
The number of elements in the symmetric difference can be expressed as:
\[
n(A \Delta B) = n(A) + n(B) - 2n(A \cap B)
\]
where \( n(A) \) and \( n(B) \) are the number of elements in sets \( A \) and \( B \), respectively, and \( n(A \cap B) \) is the number of elements common to both sets.
3. **Given Values**:
From the problem, we know:
\[
n(A) = 15 \quad \text{and} \quad n(B) = 25
\]
4. **Finding the Range of \( n(A \cap B) \)**:
The maximum number of elements in the intersection \( n(A \cap B) \) can be at most the smaller of the two sets:
\[
n(A \cap B) \leq \min(n(A), n(B)) = \min(15, 25) = 15
\]
The minimum number of elements in the intersection can be 0:
\[
n(A \cap B) \geq 0
\]
5. **Calculating Minimum and Maximum of \( n(A \Delta B) \)**:
- **Minimum**:
If \( n(A \cap B) = 15 \):
\[
n(A \Delta B) = 15 + 25 - 2 \times 15 = 10
\]
- **Maximum**:
If \( n(A \cap B) = 0 \):
\[
n(A \Delta B) = 15 + 25 - 2 \times 0 = 40
\]
6. **Range of \( n(A \Delta B) \)**:
Therefore, the range of \( n(A \Delta B) \) is:
\[
10 \leq n(A \Delta B) \leq 40
\]
7. **Finding Possible Values**:
The values of \( n(A \Delta B) \) can take are all even numbers from 10 to 40:
\[
10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40
\]
8. **Counting the Values**:
The sequence starts at 10 and ends at 40 with a common difference of 2. The number of terms \( n \) in this arithmetic sequence can be calculated using:
\[
n = \frac{\text{Last term} - \text{First term}}{\text{Common difference}} + 1
\]
Substituting the values:
\[
n = \frac{40 - 10}{2} + 1 = \frac{30}{2} + 1 = 15 + 1 = 16
\]
### Final Answer:
The number of elements in the range of \( n(A \Delta B) \) is \( \boxed{16} \).
