In a class of 95 students, each student plays at least one of hockey, cricket, and football. If 55 play hockey, 60 play cricket and 65 play football, the number of students who play all the three games could be at most

We are given that in a class of 95 students, each student plays at least one of three sports: hockey, cricket, or football. Additionally, the number of students who play each sport is:

- 55 students play hockey.
- 60 students play cricket.
- 65 students play football.

We are asked to find the **maximum number of students** who play **all three games**.

### Approach

Let:

- \( n(H) = 55 \) be the number of students who play hockey.
- \( n(C) = 60 \) be the number of students who play cricket.
- \( n(F) = 65 \) be the number of students who play football.
- \( x \) be the number of students who play all three games.
- \( n(H \cup C \cup F) = 95 \) is the total number of students, each playing at least one of the sports.

Using the principle of inclusion-exclusion for three sets, the formula for the total number of students is:

\[
n(H \cup C \cup F) = n(H) + n(C) + n(F) - n(H \cap C) - n(C \cap F) - n(F \cap H) + n(H \cap C \cap F)
\]

Substituting the known values:

\[
95 = 55 + 60 + 65 - n(H \cap C) - n(C \cap F) - n(F \cap H) + x
\]

This simplifies to:

\[
95 = 180 - n(H \cap C) - n(C \cap F) - n(F \cap H) + x
\]

\[
n(H \cap C) + n(C \cap F) + n(F \cap H) = 85 + x
\]

To **maximize** \( x \), we need to **minimize** the number of students who play exactly two sports. This occurs when the overlaps \( n(H \cap C) \), \( n(C \cap F) \), and \( n(F \cap H) \) are as small as possible.

Since the total number of students who play at least one sport is 95, and \( n(H) + n(C) + n(F) = 180 \), the maximum value of \( x \), the number of students playing all three sports, is given when the pairwise overlaps are minimized. The solution to this setup is:

\[
x = 25
\]

Thus, the maximum number of students who play all three sports is **25**.

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Would you like more details on any of the steps or concepts used?

### Related Questions:
1. How would the answer change if there were 100 students in total?
2. What if 10 students played no sports at all? How would that affect the calculation?
3. Can you explain how the inclusion-exclusion principle works in set theory?
4. What is the minimum number of students who could play all three sports?
5. How can Venn diagrams be used to

In a class of 95 students, each student plays at least one of hockey, cricket, and football. If 55 play hockey, 60 play cricket, and 65 play football, the number of students who play all three games could be at most?

Find the maximum number of students who play all three sports—hockey, cricket, and football—in a class of 95 students. Using the inclusion-exclusion principle, this problem covers key set theory concepts and mathematical optimization, making it suitable for Grades 10-12.

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