This problem can be solved using the principle of inclusion and exclusion from set theory. Let's break it down.

Definitions:

• Total students: $n = 100$
• Students playing basketball: $|B| = 80$
• Students playing chess: $|C| = 72$
• Students playing football: $|F| = 60$

We are tasked with finding the smallest possible number of students who play all three sports: $|B \cap C \cap F|$.

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Inclusion-Exclusion Principle:

For three sets $B, C,$ and $F$:

$|B \cup C \cup F| = |B| + |C| + |F| - |B \cap C| - |B \cap F| - |C \cap F| + |B \cap C \cap F|$

Since there are 100 students total, $|B \cup C \cup F| = 100$. Rearrange the equation to isolate $|B \cap C \cap F|$:

$|B \cap C \cap F| = |B| + |C| + |F| - |B \cap C| - |B \cap F| - |C \cap F| - |B \cup C \cup F|$

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Constraints:

1. $|B \cap C|, |B \cap F|, |C \cap F|$ must be at least as large as needed to fit within $|B|, |C|,$ and $|F|$.
2. To minimize $|B \cap C \cap F|$, we maximize $|B \cap C|, |B \cap F|, |C \cap F|$ within the limits of total students and overlapping sets.

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Worst-Case Scenario (Minimizing Triple Overlap):

To minimize $|B \cap C \cap F|$, assume:

1. All possible overlaps (two-sport players) are maximized to reduce the count of students required to play all three.
2. Use the constraint $|B| + |C| + |F| = 80 + 72 + 60 = 212$, which is far greater than 100. This indicates significant overlap.

To achieve the minimum: $|B \cap C \cap F| = |B| + |C| + |F| - 2 \times 100 = 212 - 200 = 12$

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Final Answer:

The smallest number of students who play all three sports is 12.

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Would you like further clarification or details? Here are 5 related questions to consider:

1. How can we calculate the largest number of students who play all three sports?
2. How would the problem change if there were more students (e.g., 150 in total)?
3. Can we visually represent this problem using a Venn diagram?
4. What if a fourth sport was introduced? How would that change the calculation?
5. How does the inclusion-exclusion principle generalize to more sets?

Tip: Practice with Venn diagrams to better understand overlapping set problems!