Answer questions based on the Venn diagram showing student preferences for skiing, volleyball, and soccer.

Let’s solve each question step by step using the Venn diagram numbers.

We have three circles: Skiing, Volleyball, Soccer.

The regions are labeled with these numbers:

- Only Skiing: 7
- Only Volleyball: 12
- Only Soccer: 20
- Skiing ∩ Volleyball only (not Soccer): 14
- Skiing ∩ Soccer only (not Volleyball): 9
- Volleyball ∩ Soccer only (not Skiing): 6
- All three (Skiing ∩ Volleyball Soccer): 15
- Outside all circles (like none of them): 19

Total students = sum of all regions = 7 + 12 + 20 + 14 + 9 + 6 + 15 + 19 = let’s check:

7+12=19; 19+20=39; 39+14=53; 53+9=62; 62+6=68; 68+15=83; 83+19=102 → Total = 102 students.

Now answer each question:

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1) How many students do not like either Skiing or Volleyball?

That means: NOT in Skiing AND NOT in Volleyball → so only in Soccer or outside both.

Look at regions that are NOT in Skiing and NOT in Volleyball:

- Only Soccer: 20
- Outside all: 19
→ 20 + 19 = 39

✔ Check: Yes — if you’re not in Skiing or Volleyball, you can only be in Soccer-only or nowhere.

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2) How many students like Volleyball or Soccer?

“Or” means union: anyone in Volleyball OR Soccer (or both).

Add all regions inside Volleyball circle OR Soccer circle.

Volleyball circle includes:
- Only Volleyball: 12
- Skiing∩Volleyball only: 14
- Volleyball∩Soccer only: 6
- All three: 15
→ 12+14+6+15 = 47

Soccer circle includes:
- Only Soccer: 20
- Skiing∩Soccer only: 9
- Volleyball∩Soccer only: 6
- All three: 15
→ 20+9+6+15 = 50

But if we add 47 + 50, we double-count the overlap (regions in both Volleyball and Soccer).

Better way: List all unique regions in Volleyball OR Soccer:

- Only Volleyball: 12
- Only Soccer: 20
- Skiing∩Volleyball only: 14
- Skiing∩Soccer only: 9
- Volleyball∩Soccer only: 6
- All three: 15
→ Add: 12+20=32; +14=46; +9=55; +6=61; +15=76

Also, is there any region missing? What about “only Skiing”? No — that’s not in Volleyball or Soccer. And outside? Not included.

So total = 76

Alternative: Total students minus those who like NEITHER Volleyball nor Soccer.

Who likes neither? Only in Skiing-only or outside.

Only Skiing: 7
Outside: 19
→ 7+19=26

Then 102 - 26 = 76 ✔ Same answer.

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3) How many students like both Skiing and Soccer but not Volleyball?

That’s the region: Skiing ∩ Soccer, but NOT in Volleyball → which is labeled 9

✔ Direct from diagram.

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4) How many students only like Skiing?

That’s the part of Skiing circle not overlapping with anything else → labeled 7

✔ Direct.

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5) How many students do not like either Volleyball or Soccer?

NOT in Volleyball AND NOT in Soccer → so only in Skiing-only or outside.

Regions:
- Only Skiing: 7
- Outside: 19
→ 7 + 19 = 26

✔ Same as earlier calculation for Q2 complement.

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6) How many students like Skiing or Soccer but not Volleyball?

“Skiing or Soccer” minus those who also like Volleyball.

First, find all in Skiing OR Soccer:

From before: Skiing OR Soccer = ?

List regions in Skiing OR Soccer:

- Only Skiing: 7
- Only Soccer: 20
- Skiing∩Volleyball only: 14 ← wait, this has Volleyball → exclude!
Wait — we want “but not Volleyball”, so remove ANY region that has Volleyball.

So include:

- Only Skiing: 7
- Only Soccer: 20
- Skiing∩Soccer only (not Volleyball): 9
- All three? NO — because it has Volleyball → exclude
- Volleyball∩Soccer only? Has Volleyball → exclude
- Skiing∩Volleyball only? Has Volleyball → exclude

So only: 7 (only Skiing) + 20 (only Soccer) + 9 (Skiing∩Soccer no VB) = 36

Check another way:

All in Skiing OR Soccer:
Skiing circle: 7 + 14 + 9 + 15 = 45
Soccer circle: 20 + 9 + 6 + 15 = 50
Union: 45 + 50 - (overlap: Skiing∩Soccer = 9 + 15 = 24) → 45+50-24=71

Now subtract those who ALSO like Volleyball (i.e., in all three or Skiing∩Volleyball or Volleyball∩Soccer):

In Skiing OR Soccer AND also in Volleyball:

- Skiing∩Volleyball only: 14
- Volleyball∩Soccer only: 6
- All three: 15
→ 14+6+15=35

Then 71 - 35 = 36 ✔ Same.

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7) How many students like Volleyball or Soccer but not Skiing?

Similar to above.

Regions in Volleyball OR Soccer, but NOT in Skiing.

So:

- Only Volleyball: 12
- Only Soccer: 20
- Volleyball∩Soccer only (not Skiing): 6
- All three? Has Skiing → exclude
- Skiing∩Volleyball? Has Skiing → exclude
- Skiing∩Soccer? Has Skiing → exclude

So: 12 + 20 + 6 = 38

Check:

Volleyball OR Soccer total = 76 (from Q2)

Subtract those who also like Skiing:

In Volleyball OR Soccer AND also in Skiing:

- Skiing∩Volleyball only: 14
- Skiing∩Soccer only: 9
- All three: 15
→ 14+9+15=38

Then 76 - 38 = 38 ✔ Correct.

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8) How many students like Skiing or Soccer?

This is union of Skiing and Soccer.

As calculated earlier: 71

Breakdown:

- Only Skiing: 7
- Only Soccer: 20
- Skiing∩Volleyball only: 14 ← wait, this is in Skiing, so include? But does it matter? We’re doing Skiing OR Soccer — yes, include all in Skiing even if they have Volleyball.

Actually, better to list all regions in Skiing OR Soccer:

- Only Skiing: 7
- Only Soccer: 20
- Skiing∩Volleyball only: 14
- Skiing∩Soccer only: 9
- Volleyball∩Soccer only: 6 ← this is in Soccer → include
- All three: 15
→ Now add: 7+20=27; +14=41; +9=50; +6=56; +15=71 ✔

Yes.

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9) How many students like both Skiing and Soccer?

That’s Skiing ∩ Soccer — includes those who may or may not like Volleyball.

So: Skiing∩Soccer only (not VB) + all three = 9 + 15 = 24

✔ Direct.

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10) How many students do not like both Skiing and Volleyball?

“Do not like both” means: NOT (Skiing AND Volleyball)

Which is everyone EXCEPT those who like BOTH Skiing and Volleyball.

Who likes both Skiing and Volleyball?

- Skiing∩Volleyball only: 14
- All three: 15
→ 14 + 15 = 29

Total students: 102

So 102 - 29 = 73

Alternatively: “do not like both” could be misinterpreted, but in logic, “not (A and B)” is correct here.

Sometimes people confuse with “like neither”, but the question says “do not like both”, which means they might like one, or neither, just not both together.

So yes, 102 - 29 = 73.

✔ Confirmed.

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Final Answers:

1) 39
2) 76
3) 9
4) 7
5) 26
6) 36
7) 38
8) 71
9) 24
10) 73

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Final Answer:
1) 39
2) 76
3) 9
4) 7
5) 26
6) 36
7) 38
8) 71
9) 24
10) 73