If X and Y are two sets such that `XuuY` has 17 elements , X has 7 elements and Y has 15 elements , how many elements does `XcapY` have ?

Answer

To find the number of elements in the intersection of sets \(X\) and \(Y\), we can use the principle of inclusion-exclusion, which states:

\[ n(X \cup Y) = n(X) + n(Y) - n(X \cap Y) \]

Where:
- \(n(X \cup Y)\) is the number of elements in the union of sets \(X\) and \(Y\),
- \(n(X)\) is the number of elements in set \(X\),
- \(n(Y)\) is the number of elements in set \(Y\),
- \(n(X \cap Y)\) is the number of elements in the intersection of sets \(X\) and \(Y\).

### Step-by-step Solution:

1. **Identify the given values:**
- \(n(X \cup Y) = 17\) (the number of elements in the union of sets \(X\) and \(Y\))
- \(n(X) = 7\) (the number of elements in set \(X\))
- \(n(Y) = 15\) (the number of elements in set \(Y\))

2. **Substitute the values into the formula:**
\[ 17 = 7 + 15 - n(X \cap Y) \]

3. **Calculate the sum of \(n(X)\) and \(n(Y)\):**
\[ 7 + 15 = 22 \]

4. **Rewrite the equation:**
\[ 17 = 22 - n(X \cap Y) \]

5. **Rearrange to solve for \(n(X \cap Y)\):**
\[ n(X \cap Y) = 22 - 17 \]

6. **Perform the subtraction:**
\[ n(X \cap Y) = 5 \]

### Conclusion:
The number of elements in the intersection of sets \(X\) and \(Y\) is \(5\).