To solve this problem, we’ll use the principle of inclusion-exclusion to determine how many students played at least one of the three sports.

Let:

• $B$ = the set of students who play basketball,
• $V$ = the set of students who play volleyball,
• $S$ = the set of students who play soccer.

We’re given:

• $|B| = 21$
• $|V| = 18$
• $|S| = 18$
• $|B \cap V| = 3$ (students who played only basketball and volleyball)
• $|B \cap S| = 2$ (students who played only basketball and soccer)
• $|V \cap S| = 4$ (students who played only volleyball and soccer)
• $|B \cap V \cap S| = 4$ (students who played all three sports).

Using Inclusion-Exclusion Principle

The formula for the number of students who played at least one of the three sports is: $|B \cup V \cup S| = |B| + |V| + |S| - |B \cap V| - |B \cap S| - |V \cap S| + |B \cap V \cap S|$

Plugging in the values: $|B \cup V \cup S| = 21 + 18 + 18 - 3 - 2 - 4 + 4$

Calculating step-by-step:

1. Sum of individual groups: $21 + 18 + 18 = 57$
2. Subtract pairwise intersections: $57 - 3 - 2 - 4 = 48$
3. Add the intersection of all three: $48 + 4 = 52$

Conclusion

Therefore, 52 students played one or more of the three sports.

Would you like a detailed breakdown of any part of this calculation, or do you have further questions?

Here are 5 related questions to explore further:

1. How many students played only one sport?
2. How many students played exactly two sports?
3. What percentage of the total students participated in all three sports?
4. If another sport was added, how would the inclusion-exclusion principle be applied?
5. How can we use Venn diagrams to represent and solve such problems?

Tip: When working with multiple overlapping groups, the inclusion-exclusion principle is a powerful tool for accurate counting without double-counting overlapping members.