n a survey of 100 students, it was found that 40 studied Mathematics, 64 
studied Physics, 35 studied Chemistry, 1 studied all the three subjects, 25 
studied Mathematics and Physics, 3 studied Mathematics and Chemistry, 20 
studied Physics and Chemistry. Use the principle of inclusion and exclusion, 
find the number of students who studied Chemistry only and the number 
who studied none of these subjects

Let's solve this problem using the **principle of inclusion and exclusion** step by step.

We define the following:

- \( n(M) = 40 \): Number of students who studied Mathematics.
- \( n(P) = 64 \): Number of students who studied Physics.
- \( n(C) = 35 \): Number of students who studied Chemistry.
- \( n(M \cap P \cap C) = 1 \): Number of students who studied all three subjects (Mathematics, Physics, and Chemistry).
- \( n(M \cap P) = 25 \): Number of students who studied both Mathematics and Physics.
- \( n(M \cap C) = 3 \): Number of students who studied both Mathematics and Chemistry.
- \( n(P \cap C) = 20 \): Number of students who studied both Physics and Chemistry.

### 1. Find the number of students who studied **only Chemistry**.

We can use the principle of inclusion and exclusion to find the number of students who studied Chemistry only:

\[
n(C \text{ only}) = n(C) - \left[ n(M \cap C) + n(P \cap C) - n(M \cap P \cap C) \right]
\]

Substituting the known values:

\[
n(C \text{ only}) = 35 - \left[ 3 + 20 - 1 \right]
\]

\[
n(C \text{ only}) = 35 - 22 = 13
\]

So, **13 students** studied Chemistry only.

### 2. Find the number of students who studied **none of these subjects**.

To find the number of students who studied none of these subjects, we first calculate the total number of students who studied at least one of the subjects using the principle of inclusion and exclusion.

\[
n(M \cup P \cup C) = n(M) + n(P) + n(C) - n(M \cap P) - n(M \cap C) - n(P \cap C) + n(M \cap P \cap C)
\]

Substitute the known values:

\[
n(M \cup P \cup C) = 40 + 64 + 35 - 25 - 3 - 20 + 1
\]

\[
n(M \cup P \cup C) = 139 - 48 = 91
\]

So, **91 students** studied at least one of the subjects.

The number of students who studied none of the subjects is:

\[
n(\text{none}) = 100 - n(M \cup P \cup C)
\]

\[
n(\text{none}) = 100 - 91 = 9
\]

Thus, **9 students** studied none of these subjects.

### Summary:
- **13 students studied Chemistry only.**
- **9 students studied none of the subjects.**

Would you like more details or have any other questions?

---

Here are 5 related questions:
1. How many students studied only Mathematics?
2. How many students studied only Physics?
3. How many students studied exactly two subjects?
4. How many students studied both Mathematics and Chemistry, but not Physics?
5. How would the result change if one more student studied all three subjects?

**Tip**: Always double-check your inclusion-exclusion calculations by verifying each group separately.

In a survey of 100 students, it was found that 40 studied Mathematics, 64 studied Physics, 35 studied Chemistry, 1 studied all the three subjects, 25 studied Mathematics and Physics, 3 studied Mathematics and Chemistry, 20 studied Physics and Chemistry. Use the principle of inclusion and exclusion, find the number of students who studied Chemistry only and the number who studied none of these subjects.

Solve a problem involving 100 students where they studied Mathematics, Physics, and Chemistry using the principle of inclusion and exclusion. This detailed solution shows how to find the number of students who studied only Chemistry and those who studied none of the subjects. Suitable for Grades 10-12 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation