In a class of 31 students 22 study Physics 20 study chemistry and five students study neither calculate the number of students who study both subject

Let:
- \( P \) represent the set of students who study Physics.
- \( C \) represent the set of students who study Chemistry.

We have:
1. \( |P| = 22 \) (students studying Physics),
2. \( |C| = 20 \) (students studying Chemistry),
3. \( |P \cup C| = 31 - 5 = 26 \) (students studying at least one subject).

Using the principle of inclusion-exclusion:
\[
|P \cup C| = |P| + |C| - |P \cap C|
\]
Substitute the known values:
\[
26 = 22 + 20 - |P \cap C|
\]
\[
26 = 42 - |P \cap C|
\]
\[
|P \cap C| = 42 - 26 = 16
\]

**Answer:** 16 students study both subjects.

Would you like a breakdown of this solution? 

Here are 5 related questions:
1. What if 10 students studied neither subject? How would that affect the solution?
2. How many students study only Physics?
3. How many students study only Chemistry?
4. How would we approach this if we knew students studying exactly one subject?
5. How does the principle of inclusion-exclusion apply to more than two sets?

**Tip:** When using the inclusion-exclusion principle, always start by defining each set clearly to avoid confusion.

In a class of 31 students 22 study Physics 20 study chemistry and five students study neither calculate the number of students who study both subjects.

Learn to solve a class distribution problem using the inclusion-exclusion principle. Given 31 students with 22 studying Physics, 20 studying Chemistry, and 5 studying neither, find the number who study both subjects. Suitable for Grades 9-11.

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