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The problem describes a survey of 215 teachers, and it gives details about the number of teachers teaching different subjects (Maths, Physics, and Computer Studies) with some overlaps between these subjects.

Let's first summarize the key details from the problem:

- Total number of teachers surveyed: **215**
- Teachers who taught **Maths**: **170**
- Teachers who taught **Physics**: **100**
- Teachers who taught **Computer Studies**: **141**
- Teachers who taught both **Maths and Physics**: **65**
- Teachers who taught both **Physics and Computer Studies**: **30**
- Teachers who taught both **Computer Studies and Maths**: **26**
- Teachers who taught all three subjects: **21**

### Part (i): Illustrate the information in a Venn diagram

We have three subjects: Maths (M), Physics (P), and Computer Studies (C). The Venn diagram will include three intersecting circles representing these subjects, with numbers placed in the respective intersections based on the data provided.

Let's break down the numbers for each region:
- Teachers who taught **all three subjects (M ∩ P ∩ C)**: 21
- Teachers who taught **Maths and Physics only**: This is the overlap of Maths and Physics, excluding those who also teach Computer Studies. So, this number is \( 65 - 21 = 44 \).
- Teachers who taught **Physics and Computer Studies only**: This is the overlap of Physics and Computer Studies, excluding those who also teach Maths. So, this number is \( 30 - 21 = 9 \).
- Teachers who taught **Maths and Computer Studies only**: This is the overlap of Maths and Computer Studies, excluding those who also teach Physics. So, this number is \( 26 - 21 = 5 \).
- Teachers who taught **Maths only**: This is the total number of Maths teachers minus those who also teach Physics or Computer Studies or both. So, \( 170 - (44 + 21 + 5) = 100 \).
- Teachers who taught **Physics only**: This is the total number of Physics teachers minus those who also teach Maths or Computer Studies or both. So, \( 100 - (44 + 21 + 9) = 26 \).
- Teachers who taught **Computer Studies only**: This is the total number of Computer Studies teachers minus those who also teach Maths or Physics or both. So, \( 141 - (5 + 21 + 9) = 106 \).

Once you fill out the Venn diagram with these numbers, you'll have a clear illustration.

### Part (ii): How many teachers taught none of the subjects?

To find this, we subtract the total number of teachers involved in teaching at least one subject from the total surveyed:
- Total number of teachers who teach at least one subject = sum of all areas in the Venn diagram:
  \( 100 + 26 + 106 + 44 + 9 + 5 + 21 = 311 \).

Since the total surveyed is 215, it seems we need to adjust the question based on those provided (there may be a slight inconsistency or overlap).

Would you like further explanation or help drawing the diagram?

Here are some related questions:
1. How would adding another subject change the Venn diagram?
2. Can you calculate the percentage of teachers teaching only one subject?
3. What is the significance of those teaching all three subjects?
4. How would this data change if 10 more teachers were added to each category?
5. What kind of assumptions are made with this type of survey data?

**Tip**: When working with Venn diagrams, always start with the innermost intersections (those teaching all three) and work outward to avoid double-counting.

A survey carried out in a certain district showed that out of 215 teachers, 170 taught Maths, 100 Physics, 141 Computer Studies, 65 taught Maths and Physics, 30 Physics and Computer Studies, 26 Computer Studies and Maths, and 21 all three subjects. (i) Illustrate the information in a Venn diagram. (ii) How many teachers taught none of the subjects? (iii) Teachers who taught Physics or Computer Studies but not Maths. (iv) Teachers who taught two subjects only.

This Venn diagram problem focuses on a survey of 215 teachers, analyzing how many teach Maths, Physics, and Computer Studies. Learn how to use the principle of inclusion and exclusion to answer questions about overlaps and exclusions.

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