Given: Total number of students = 50

Number of students who offered Physics = 40

Number of students who offered Biology = 30

We want to find the number of students who offered both Physics and Biology.

Let X be the set of students who offered Physics, and \(Y\) be the set of students who offered Biology.

Number of students who offered both Physics and Biology = \(|X \cap Y|\) (the intersection of sets \(X\) and \(Y\)).

Using the principle of inclusion-exclusion:

\(|X \cup Y| = |X| + |Y| - |X \cap Y|\)

Where:

\(|X \cup Y|\) is the total number of students who offered either Physics or Biology or both.

Substitute the given values:

\(50 = 40 + 30 - |X \cap Y|\)

Solving for \(|X \cap Y|\):

\(|X \cap Y| = 40 + 30 - 50 = 20\)