1. The problem asks to find the number of elements in the intersection of two sets A and B given some values. 2. Given: $n(A) = 27$, $n(B) = 25$, $n(A \cap B) = x$, and $n(A \cup B) = 30$. 3. From the Venn diagram, the number of elements only in A is $27 - x$. 4. The number of elements only in B is $25 - x$. 5. The number of elements in the intersection is $x$. 6. The union of A and B is the sum of elements only in A, only in B, and in both: $$n(A \cup B) = (27 - x) + (25 - x) + x = 27 + 25 - x = 52 - x$$ 7. We know $n(A \cup B) = 30$, so set up the equation: $$52 - x = 30$$ 8. Solve for $x$: $$x = 52 - 30 = 22$$ 9. Therefore, the number of elements in the intersection $n(A \cap B) = 22$. 10. Summary: - Only in A: $27 - 22 = 5$ - Only in B: $25 - 22 = 3$ - In both: $22$ Final answer: $x = 22$