Among the primary 6 students in a school, 60 like mathematics, 50 like English and 42 like sports. 30 of them like both mathematics and English, 20 of them like both English and sports and 35 of them like both mathematics and sports. If 18 students like all the three subjects, how many students have exactly one favourite subject?

To find how many students have exactly one favourite subject, we can use the principle of inclusion-exclusion (PIE) to calculate the total number of students first. Then we will use this total to find the number of students who like exactly one subject.

Let:
- \( M \) be the set of students who like mathematics.
- \( E \) be the set of students who like English.
- \( S \) be the set of students who like sports.

We are given:
- \(|M| = 60\)
- \(|E| = 50\)
- \(|S| = 42\)
- \(|M \cap E| = 30\)
- \(|E \cap S| = 20\)
- \(|M \cap S| = 35\)
- \(|M \cap E \cap S| = 18\)

The formula for the total number of students using PIE is:
\[
|M \cup E \cup S| = |M| + |E| + |S| - |M \cap E| - |E \cap S| - |M \cap S| + |M \cap E \cap S|
\]

Substitute the given values:
\[
|M \cup E \cup S| = 60 + 50 + 42 - 30 - 20 - 35 + 18
\]
\[
|M \cup E \cup S| = 60 + 50 + 42 - 85 + 18
\]
\[
|M \cup E \cup S| = 102 - 85 + 18
\]
\[
|M \cup E \cup S| = 35
\]

So, the total number of students is 35.

Next, we need to find the number of students who like exactly one subject. To do this, we use the following formula for each subject \(X\):

\[
\text{Number of students who like only } X = |X| - |X \cap Y| - |X \cap Z| + |X \cap Y \cap Z|
\]

For mathematics:
\[
\text{Number of students who like only mathematics} = |M| - |M \cap E| - |M \cap S| + |M \cap E \cap S|
\]
\[
= 60 - 30 - 35 + 18
\]
\[
= 60 - 65 + 18
\]
\[
= -5 + 18
\]
\[
= 13
\]

For English:
\[
\text{Number of students who like only English} = |E| - |E \cap M| - |E \cap S| + |E \cap M \cap S|
\]
\[
= 50 - 30 - 20 + 18
\]
\[
= 50 - 50 + 18
\]
\[
= 18
\]

For sports:
\[
\text{Number of students who like only sports} = |S| - |S \cap M| - |S \cap E| + |S \cap M \cap S|
\]
\[
= 42 - 35 - 20 + 18
\]
\[
= 42 - 55 + 18
\]
\[
= -13 + 18
\]
\[
= 5
\]

Summing up the students who like exactly one subject:
\[
13 + 18 + 5 = 36
\]

Thus, the number of students who have exactly one favourite subject is \( 36 \).

Would you like any further details or explanations?

Here are some related questions to expand your understanding:

1. How is the principle of inclusion-exclusion (PIE) applied in other scenarios?
2. What is the probability that a randomly selected student likes exactly one subject?
3. How can Venn diagrams be used to visualize the given problem?
4. What is the impact on the results if the number of students who like all three subjects changes?
5. How would the calculation change if there were more subjects?
6. Can this method be used to solve problems involving four or more sets?
7. What are other real-life applications of the inclusion-exclusion principle?
8. How would we calculate the number of students who like exactly two subjects?

**Tip:** Practice using Venn diagrams to better understand and solve set problems visually.

Explore how to calculate the number of students who like exactly one subject among Mathematics, English, and Sports using the Principle of Inclusion-Exclusion. Detailed solution includes formulas and step-by-step calculations.

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