If $n(A)=20$ and $n(B)=30$ and $n(A\cup B)=40$ then $n(A\cap B)$ is ________.

Step 1 (Inclusion-Exclusion Formula): The Principle of Inclusion-Exclusion (PIE) for two sets $A$ and $B$ states that:

Step 2 (Substitute the Values): We are given:

• $n(A)=20$

• $n(B)=30$

• $n(A\cup B)=40$

Substituting these values into the formula:

Step 3 (Conclusion): The number of elements in the intersection $n(A\cap B)$ is exactly 10.