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705-805 Level|   Overlapping Sets|   Remainders|         
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satish_sahoo
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In a class of 50 students, 20 play Hockey, 15 play Cricket and 11 play 03 Oct 2024, 01:03
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GMATNinja

Can this be solved using the "counting's approach" instead of venn diagram?
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In a class of 50 students, 20 play Hockey, 15 play Cricket and 11 play 04 Oct 2024, 06:31
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Hi
there is a mistake in your Venn Diagram. As per the question, 7 plays both hockey and cricket . it does not mean these 7 ONLY play hockey and cricket. there could be some of these 7 who play all three sports. Check the following Venn Diagram

add these individual blocks , sum of which must be 50
you will get m = 2
then add 7-m+4-m+5-m = 16-3m = 10
PareshGmat
Answer = 10
Using Venn Diagram
Bunuel, can you please tell if this method is correct?. Got x -ve in this case
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In a class of 50 students, 20 play Hockey, 15 play Cricket and 11 play 04 Nov 2024, 00:08
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In a class of 50 students, 20 play Hockey, 15 play Cricket and 11 play Football. 7 play both Hockey and Cricket, 4 play Cricket and Football and 5 play Hockey and football. If 18 students do not play any of these given sports, how many students play exactly two of these sports?

A. 12
B. 10
C. 11
D. 15
E. 14
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Check this post first to visualize the formula given below: https://anaprep.com/sets-statistics-thr ... ping-sets/

Total - Neither = H + C + F - (H and C + C and F + H and F) + (H and C and F)


50 - 18 = 20 + 15 + 11 - (7 + 4 + 5) + (H and C and F)

(H and C and F) = 2

Hence, of the 7 that play H and C, 2 play all 3 sports and hence 5 play only H and C.
Of the 4 who play C and F, 2 play all three sports so 2 play only C and F.
Of the 5 who play H and F, 2 play all three sports so 3 play only H and F.

So 5 + 2 + 3 = 10 play only two sports

Answer (B)
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In a class of 50 students, 20 play Hockey, 15 play Cricket and 11 play 15 Dec 2024, 12:54
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satish_sahoo
GMATNinja

Can this be solved using the "counting's approach" instead of venn diagram?
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Sorry for the slow response! The "counting approach" that you sometimes see in our videos (on question #3 of this video, for example) might not be the most efficient approach, but yes, it'll work.

The numbers are slightly different, but we also explained that approach for a nearly identical problem here: https://gmatclub.com/forum/in-a-class-o ... l#p3434914. Try applying that to this one, and let us know if you still have questions?
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In a class of 50 students, 20 play Hockey, 15 play Cricket and 11 play 18 Dec 2024, 07:33
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Hi Scott, I could not find this formula in TTP where we are adding the number in triple overlaps. The formula in TTP suggests #H + #C + #F - #H&C - #C&F - #H&F - 2#H&C&F + Neither.

Kindly reply
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gmihir
In a class of 50 students, 20 play Hockey, 15 play Cricket and 11 play Football. 7 play both Hockey and Cricket, 4 play Cricket and Football and 5 play Hockey and football. If 18 students do not play any of these given sports, how many students play exactly two of these sports?

A. 12
B. 10
C. 11
D. 15
E. 14

Since 18 students do not play any of the three sports, 50 - 18 = 32 students must play at least one of the 3 sports. This total can be formulated as follows:

Total = #(H) + #(C) + #(F) - #(H and C) - #(C and F) - #(H and F) + #(H and C and F)

Thus, we have:

32 = 20 + 15 + 11 - 7 - 4 - 5 + #(H and C and F)

32 = 30 + #(H and C and F)

2 = #(H and C and F)

Since #(H and C) = 7 (which also include those who play Football), but we’ve found that #(H and C and F) = 2, there must be 7 - 2 = 5 students who play Hockey and Cricket only. Similarly, there must be 4 - 2 = 2 students who play Cricket and Football only, and 5 - 2 = 3 students who play Hockey and Football only. Thus, there must be 5 + 2 + 3 = 10 students who play exactly 2 sports.

Answer: B
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In a class of 50 students, 20 play Hockey, 15 play Cricket and 11 play 16 Mar 2025, 08:47
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Bunuel I think there's a mistake in your formula,

Is the formula not,

Total - Neither = A + B + C - (exactly 2) - 2(all 3) ?

Please clarify this.

KarishmaB chetan2u
Bunuel
gmihir
In a class of 50 students, 20 play Hockey, 15 play Cricket and 11 play Football. 7 play both Hockey and Cricket, 4 play Cricket and Football and 5 play Hockey and football. If 18 students do not play any of these given sports, how many students play exactly two of these sports?

A. 12
B. 10
C. 11
D. 15
E. 14

Notice that "7 play both Hockey and Cricket" does not mean that out of those 7, some does not play Football too. The same for Cricket/Football and Hockey/Football.

\(\{Total\} = \{Hockey\} + \{Cricket\} + \{Football\} - \{HC + HF + CF\} + \{All \ three\} + \{Neither\}\)
(For more check ADVANCED OVERLAPPING SETS PROBLEMS)

\(50 = 20 + 15 + 11 -(7 + 4 + 5) + \{All \ three\} + 18\);
\(\{All \ three\}=2\);

Those who play ONLY Hockey and Cricket are 7 - 2 = 5;
Those who play ONLY Cricket and Football are 4 - 2 = 2;
Those who play ONLY Hockey and Football are 5 - 2 = 3;

Hence, 5 + 2 + 3 = 10 students play exactly two of these sports.

Answer: B.
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In a class of 50 students, 20 play Hockey, 15 play Cricket and 11 play 16 Mar 2025, 09:28
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Bunuel I think there's a mistake in your formula,

Is the formula not,

Total - Neither = A + B + C - (exactly 2) - 2(all 3) ?

Please clarify this.

KarishmaB chetan2u
Bunuel
gmihir
In a class of 50 students, 20 play Hockey, 15 play Cricket and 11 play Football. 7 play both Hockey and Cricket, 4 play Cricket and Football and 5 play Hockey and football. If 18 students do not play any of these given sports, how many students play exactly two of these sports?

A. 12
B. 10
C. 11
D. 15
E. 14

Notice that "7 play both Hockey and Cricket" does not mean that out of those 7, some does not play Football too. The same for Cricket/Football and Hockey/Football.

\(\{Total\} = \{Hockey\} + \{Cricket\} + \{Football\} - \{HC + HF + CF\} + \{All \ three\} + \{Neither\}\)
(For more check ADVANCED OVERLAPPING SETS PROBLEMS)

\(50 = 20 + 15 + 11 -(7 + 4 + 5) + \{All \ three\} + 18\);
\(\{All \ three\}=2\);

Those who play ONLY Hockey and Cricket are 7 - 2 = 5;
Those who play ONLY Cricket and Football are 4 - 2 = 2;
Those who play ONLY Hockey and Football are 5 - 2 = 3;

Hence, 5 + 2 + 3 = 10 students play exactly two of these sports.

Answer: B.
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The data given to us in the question is not for people playing exactly two sports. That's the reason we use: Total - Neither = H + C + F - {Sum of people playing 2 sports) + people playing 3 sports.

From here, we find the number of people playing 3 sports (which is 2) and then subtract that number from each of 2 sports (HC - 2, HF - 2, CF - 2), the sum of which will then give us number of people playing exactly two sports.
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In a class of 50 students, 20 play Hockey, 15 play Cricket and 11 play 16 Mar 2025, 19:17
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Sujithz001
Bunuel I think there's a mistake in your formula,

Is the formula not,

Total - Neither = A + B + C - (exactly 2) - 2(all 3) ?

Please clarify this.

KarishmaB chetan2u
Bunuel
gmihir
In a class of 50 students, 20 play Hockey, 15 play Cricket and 11 play Football. 7 play both Hockey and Cricket, 4 play Cricket and Football and 5 play Hockey and football. If 18 students do not play any of these given sports, how many students play exactly two of these sports?

A. 12
B. 10
C. 11
D. 15
E. 14

Notice that "7 play both Hockey and Cricket" does not mean that out of those 7, some does not play Football too. The same for Cricket/Football and Hockey/Football.

\(\{Total\} = \{Hockey\} + \{Cricket\} + \{Football\} - \{HC + HF + CF\} + \{All \ three\} + \{Neither\}\)
(For more check ADVANCED OVERLAPPING SETS PROBLEMS)

\(50 = 20 + 15 + 11 -(7 + 4 + 5) + \{All \ three\} + 18\);
\(\{All \ three\}=2\);

Those who play ONLY Hockey and Cricket are 7 - 2 = 5;
Those who play ONLY Cricket and Football are 4 - 2 = 2;
Those who play ONLY Hockey and Football are 5 - 2 = 3;

Hence, 5 + 2 + 3 = 10 students play exactly two of these sports.

Answer: B.
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Hi

The formula depends on what set of people/things are included in (twos).
If it is exactly two, then it means we have not subtracted all three and so subtract all three two times as it got added thrice when we added A, B and C.
If it is two and thus includes all three too, the portion of all three gets subtracted thrice while we should have subtracted only twice, so added it once back .

Venn Diagram would help you visualize it better.
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In a class of 50 students, 20 play Hockey, 15 play Cricket and 11 play 16 Mar 2025, 21:20
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Sujithz001
Bunuel I think there's a mistake in your formula,

Is the formula not,

Total - Neither = A + B + C - (exactly 2) - 2(all 3) ?

Please clarify this.

KarishmaB chetan2u
Bunuel
gmihir
In a class of 50 students, 20 play Hockey, 15 play Cricket and 11 play Football. 7 play both Hockey and Cricket, 4 play Cricket and Football and 5 play Hockey and football. If 18 students do not play any of these given sports, how many students play exactly two of these sports?

A. 12
B. 10
C. 11
D. 15
E. 14

Notice that "7 play both Hockey and Cricket" does not mean that out of those 7, some does not play Football too. The same for Cricket/Football and Hockey/Football.

\(\{Total\} = \{Hockey\} + \{Cricket\} + \{Football\} - \{HC + HF + CF\} + \{All \ three\} + \{Neither\}\)
(For more check ADVANCED OVERLAPPING SETS PROBLEMS)

\(50 = 20 + 15 + 11 -(7 + 4 + 5) + \{All \ three\} + 18\);
\(\{All \ three\}=2\);

Those who play ONLY Hockey and Cricket are 7 - 2 = 5;
Those who play ONLY Cricket and Football are 4 - 2 = 2;
Those who play ONLY Hockey and Football are 5 - 2 = 3;

Hence, 5 + 2 + 3 = 10 students play exactly two of these sports.

Answer: B.
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Sujithz001 - :) :) I would think 10 times before writing those words: "Bunuel I think there's a mistake in your formula" ...and perhaps still not write them! I would think I need to go to sleep. :) :) But that is something we all learn with time, don't worry.

There are different formulas for different contexts. It depends on what is given to us.

This is correct:
Total - Neither = A + B + C - (exactly 2) - 2(All 3)

When I have data available for how many people are in exactly 2 sets, I can use this formula.

This is correct too:
Total - Neither = A + B + C - (A and B + B and C + C and A) + All 3

Depends on what you are subtracting and what you are adding back later. I have explained how many such formulas are inter-related using Venn diagrams in my overlapping sets module. You can check it out using the 3 day free trial.
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In a class of 50 students, 20 play Hockey, 15 play Cricket and 11 play 29 Sep 2025, 09:30
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This is a classic three-set overlapping problem - these can definitely be tricky until you see the key insight. Let me walk you through this step-by-step.

The Key Insight You Need:
When the problem says "7 play both Hockey and Cricket," this includes students who might also play Football! This is where most students get confused. Let's think about this systematically.

Step 1: Set Up Your Known Values
Let's organize what we know:
- Total students = 50
- Students playing NO sports = 18
- Therefore, students playing at least one sport = \(50 - 18 = 32\)

Individual sports:
- Hockey (H) = 20
- Cricket (C) = 15
- Football (F) = 11

Overlaps (at least two sports):
- H and C = 7
- C and F = 4
- H and F = 5
- All three = ? (let's call this \(x\))

Step 2: Use Inclusion-Exclusion to Find Students Playing All Three
Here's what you need to see - when we add all the individual sport players (\(20 + 15 + 11 = 46\)), we're counting some students multiple times.

Using the inclusion-exclusion formula:
\(|H \cup C \cup F| = |H| + |C| + |F| - |H \cap C| - |C \cap F| - |H \cap F| + |H \cap C \cap F|\)

Substituting our values:
\(32 = 20 + 15 + 11 - 7 - 4 - 5 + x\)
\(32 = 46 - 16 + x\)
\(32 = 30 + x\)
\(x = 2\)

So 2 students play all three sports!

Step 3: Calculate Exactly Two Sports
Now here's the crucial part - to find students playing exactly two sports, we subtract those playing all three from each pair:

- Exactly Hockey and Cricket (not Football): \(7 - 2 = 5\)
- Exactly Cricket and Football (not Hockey): \(4 - 2 = 2\)
- Exactly Hockey and Football (not Cricket): \(5 - 2 = 3\)

Total playing exactly two sports: \(5 + 2 + 3 = 10\)

Answer: B (10 students)

Notice how the trick was recognizing that the given overlaps included the "all three" students? That's the pattern you want to watch for in these problems.

You can check out the step-by-step solution on Neuron by e-GMAT to master the inclusion-exclusion principle systematically and learn a powerful verification technique that ensures you never make calculation errors. You can also explore other GMAT official questions with detailed solutions on Neuron for structured practice here.
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