There are 100 students in a particular class, 60% students play cricket, 30% students play football, and 10% students play both the games. What is the number of students who play neither cricket nor football?

To solve this problem, we can use the principle of inclusion-exclusion. Let's denote the following:

- \( C \) as the set of students who play cricket.
- \( F \) as the set of students who play football.
- \( |C| \) as the number of students who play cricket.
- \( |F| \) as the number of students who play football.
- \( |C \cap F| \) as the number of students who play both cricket and football.

Given:

- Total number of students = 100
- \( |C| = 60\% \) of 100 = 60 students
- \( |F| = 30\% \) of 100 = 30 students
- \( |C \cap F| = 10\% \) of 100 = 10 students

Using the principle of inclusion-exclusion, the number of students who play either cricket or football (or both) is:
\[
|C \cup F| = |C| + |F| - |C \cap F|
\]
Substituting the values:
\[
|C \cup F| = 60 + 30 - 10 = 80
\]
Thus, 80 students play either cricket, football, or both.

The number of students who play neither cricket nor football is:
\[
\text{Students who play neither} = \text{Total students} - |C \cup F| = 100 - 80 = 20
\]

Therefore, 20 students play neither cricket nor football.

Would you like further details or have any more questions?

Here are 5 related questions for further practice:

1. If 40 students play basketball in addition to cricket and football, how many students play only basketball?
2. If 5 students play all three games: cricket, football, and basketball, how many students play exactly two of these games?
3. If the percentage of students playing both cricket and football is increased to 15%, how does this change the number of students who play neither game?
4. What percentage of students play only one of the two games (cricket or football)?
5. If the total number of students increases to 120, how does the number of students who play neither game change?

**Tip:** The principle of inclusion-exclusion is a powerful tool to avoid overcounting when dealing with overlapping sets.

Learn how to solve the problem of determining the number of students who play neither cricket nor football using the principle of inclusion-exclusion. This problem involves set theory and percentages, suitable for high school students.

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