How to Solve Venn Diagrams with 3 Circles - mathsathome.com - Free Printable

Looking at the problem:

We are told that 30 students were asked which sports they play.

We’re given:
- 20 play basketball (B)
- 16 play football (F)
- 15 play tennis (T)
- 10 play basketball and tennis
- 11 play basketball and football
- 9 play football and tennis
- 7 play all three

The Venn diagram shows a yellow region labeled “7” in the center — that’s the intersection of all three circles, meaning 7 students play all three sports.

But wait — the line “7 play all three” is crossed out in the image. That might be a hint that we need to find how many play all three, or perhaps it’s indicating that this number is already known and used? Actually, looking again — the yellow region is labeled “7”, so likely the problem is showing us that 7 is the number for all three, and maybe we’re being asked to verify or use that?

Actually, re-reading the user’s request: they just uploaded the image and said “solve the problem”. But the image doesn’t explicitly state what question to answer. However, in typical Venn diagram problems like this, when you’re given totals for each set, pairwise intersections, and sometimes the triple intersection, the usual question is: How many students play none of the sports? Or sometimes: Fill in the Venn diagram.

But since the image has the “7” already filled in the center, and the text says “7 play all three” (even though it’s crossed out — maybe that’s a red herring or correction?), let’s assume the task is to find how many students play none of the three sports.

That’s a common question with these setups.

So, let’s solve: How many students play at least one sport? Then subtract from 30 to get those who play none.

We’ll use the principle of inclusion-exclusion.

Formula for union of three sets:

|B ∪ F ∪ T| = |B| + |F| + |T| - |B∩F| - |B∩T| - |F∩T| + |B∩F∩T|

Plug in the numbers:

= 20 + 16 + 15 - 11 - 10 - 9 + 7

Calculate step by step:

20 + 16 = 36
36 + 15 = 51
51 - 11 = 40
40 - 10 = 30
30 - 9 = 21
21 + 7 = 28

So, 28 students play at least one sport.

Total students = 30

Therefore, students who play none = 30 - 28 = 2

Let me double-check with another method — filling the Venn diagram regions.

Start with the center: all three = 7

Now, basketball and tennis only (not football): total basketball and tennis is 10, but that includes those who also play football. So subtract the 7 who play all three: 10 - 7 = 3

Similarly, basketball and football only: 11 - 7 = 4

Football and tennis only: 9 - 7 = 2

Now, basketball only: total basketball is 20. Subtract those who play basketball and tennis only (3), basketball and football only (4), and all three (7). So 20 - 3 - 4 - 7 = 6

Football only: total football 16. Subtract football and basketball only (4), football and tennis only (2), and all three (7). So 16 - 4 - 2 - 7 = 3

Tennis only: total tennis 15. Subtract tennis and basketball only (3), tennis and football only (2), and all three (7). So 15 - 3 - 2 - 7 = 3

Now add up all regions:

Only B: 6
Only F: 3
Only T: 3
B&F only: 4
B&T only: 3
F&T only: 2
All three: 7

Sum: 6+3+3+4+3+2+7 = let’s add:

6+3=9; +3=12; +4=16; +3=19; +2=21; +7=28

Same as before.

So 28 play at least one sport → 30 - 28 = 2 play none.

Final Answer: 2