Let `X = {1, 2, 3,.......... 10} and P = {1, 2, 3, 4, 5}`. The number of subsets Q of X such that `P Delta Q = {3)` is __________

Answer

To solve the problem, we need to find the number of subsets \( Q \) of the set \( X = \{1, 2, 3, \ldots, 10\} \) such that the symmetric difference \( P \Delta Q = \{3\} \), where \( P = \{1, 2, 3, 4, 5\} \).

### Step-by-step Solution:

1. **Understanding Symmetric Difference**:
The symmetric difference \( P \Delta Q \) is defined as:
\[ P \Delta Q = (P - Q) \cup (Q - P) \]
This means it consists of elements that are in either \( P \) or \( Q \), but not in both.

2. **Setting up the Condition**:
We need \( P \Delta Q = \{3\} \). This implies that the only element in the symmetric difference must be 3. Therefore, we can conclude:
- The element 3 must be in \( P \) but not in \( Q \).
- The elements in \( Q \) must not include 3.

3. **Identifying Elements in \( P \) and \( Q \)**:
Since \( P = \{1, 2, 3, 4, 5\} \), for \( P \Delta Q \) to equal \( \{3\} \):
- \( P - Q \) must include 3.
- \( Q - P \) must be empty, meaning \( Q \) can only contain elements from \( P \) that are not in \( P \).

4. **Elements of \( Q \)**:
Since 3 cannot be in \( Q \), the possible elements of \( Q \) are from the set \( P \) excluding 3, which gives us:
\[ P' = \{1, 2, 4, 5\} \]
Thus, \( Q \) can only consist of the elements from \( P' \).

5. **Counting Subsets**:
The set \( P' \) has 4 elements: \( \{1, 2, 4, 5\} \). The number of subsets of a set with \( n \) elements is given by \( 2^n \). Therefore, the number of subsets of \( P' \) is:
\[ 2^4 = 16 \]

6. **Conclusion**:
The total number of subsets \( Q \) of \( X \) such that \( P \Delta Q = \{3\} \) is \( 16 \).

### Final Answer:
The number of subsets \( Q \) of \( X \) such that \( P \Delta Q = \{3\} \) is **16**.

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