Question

In class of 100 students, 37 study mathematics, 32 study physics and 31 study chemistry. 9 students study mathematics and physics, 12 students study mathematics and chemistry, 11 students study physics and chemistry and 28 students study none of these subjects. 1. Illustrate the information in a Venn diagram
2. Find the number of students who study all the three subjects
3. Fine the number of students who study only mathematics, only physics and only chemistry
4. How many students study exactly two subjects ?

Answer 1

To address the problem systematically, we will define the following sets:

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

Given the data:

- Total students: \( T = 100 \)
- Students studying Mathematics: \( |M| = 37 \)
- Students studying Physics: \( |P| = 32 \)
- Students studying Chemistry: \( |C| = 31 \)
- Students studying both Mathematics and Physics: \( |M \cap P| = 9 \)
- Students studying both Mathematics and Chemistry: \( |M \cap C| = 12 \)
- Students studying both Physics and Chemistry: \( |P \cap C| = 11 \)
- Students studying none of the subjects: \( 28 \)

The remaining students who study at least one subject is:

\[
100 - 28 = 72
\]

Let \( x \) represent the number of students studying all three subjects \( |M \cap P \cap C| \).

### Step 1: Venn Diagram Illustration
A Venn diagram would show three intersecting circles labeled \( M \), \( P \), and \( C \), with overlaps that denote students taking multiple subjects, but let's calculate actual numbers before drawing.

### Step 2: Finding the Number of Students Who Study All Three Subjects
Using the principle of inclusion-exclusion:

\[
|M \cup P \cup C| = |M| + |P| + |C| - |M \cap P| - |M \cap C| - |P \cap C| + |M \cap P \cap C|
\]

Substituting in the known values:

\[
72 = 37 + 32 + 31 - 9 - 12 - 11 + x
\]

Simplifying the equation:

\[
72 = 100 - 32 + x
\]
\[
72 = 68 + x
\]
\[
x = 72 - 68 = 4
\]

### Conclusion for Step 2:
The number of students studying all three subjects is \( **4** \).

### Step 3: Finding Students Studying Only One Subject
Now that we know \( x = 4 \), we can find the number of students studying only Mathematics, only Physics, and only Chemistry.

- Only Mathematics:

\[
|M \text{ only}| = |M| - (|M \cap P| + |M \cap C| - |M \cap P \cap C|)
\]
\[
|M \text{ only}| = 37 - (9 + 12 - 4) = 37 - 17 = 20
\]

- Only Physics:

\[
|P \text{ only}| = |P| - (|M \cap P| + |P \cap C| - |M \cap P \cap C|)
\]
\[
|P \text{ only}| = 32 - (9 + 11 - 4) = 32 - 16 = 16
\]

- Only Chemistry:

\[
|C \text{ only}| = |C| - (|M \cap C| + |P \cap C| - |M \cap P \cap C|)
\]
\[
|C \text{ only}| = 31 - (12 + 11 - 4) = 31 - 19 = 12
\]

### Conclusion for Step 3:
- Number of students studying only Mathematics: \( **20** \)
- Number of students studying only Physics: \( **16** \)
- Number of students studying only Chemistry: \( **12** \)

### Step 4: Finding Students Studying Exactly Two Subjects
Students studying exactly two subjects can be calculated as:

- Mathematics and Physics only:

\[
|M \cap P \text{ only}| = |M \cap P| - |M \cap P \cap C| = 9 - 4 = 5
\]

- Mathematics and Chemistry only:

\[
|M \cap C \text{ only}| = |M \cap C| - |M \cap P \cap C| = 12 - 4 = 8
\]

- Physics and Chemistry only:

\[
|P \cap C \text{ only}| = |P \cap C| - |M \cap P \cap C| = 11 - 4 = 7
\]

Adding these gives the total number of students who study exactly two subjects:

\[
|M \cap P \text{ only}| + |M \cap C \text{ only}| + |P \cap C \text{ only}| = 5 + 8 + 7 = 20
\]

### Conclusion for Step 4:
The number of students studying exactly two subjects is \( **20** \).

### Summary of Results:
1. Venn Diagram can be drawn using the set values calculated.
2. Students studying all three subjects: \( **4** \)
3. Students studying only Mathematics: \( **20** \), only Physics: \( **16** \), only Chemistry: \( **12** \)
4. Students studying exactly two subjects: \( **20** \)