The answer is 10. Not 13.
As stated in an earlier post, the formula offered by instructor Wolff is unfortunately incorrect. Using it leads to a contradiction as demonstrated in the video I uploaded. Solving out the Venn using her method leaves one person unaccounted for. The video also shows a loose proof of the correct formula. Please watch if you you are not convinced.

https://www.youtube.com/watch?v=SM_ABQXScMw

Finally, note the work of MM_Ed, ma127, patanjali and pinchharmonic at various points concur with my conclusions.
I feel bad for having to point out this error again, but it is an error and the formula is not reliable as written. Please ask another expert for confirmation.

Hi MrR,
I agree. I solved it using the Venn and algebra see below:



Now as there are 18 non players, therefore, there are 32 total players.
So:
(8+x)+(4+x)+(2+x)+(7-x)+(5-x)+(4-x)+(x) [there are 3 (-x) and 4 (+x) so, in sum only 1 x]
> 8+7+5+2+4+4+x = 32
> x = 32-30
> x = 2

So now to calculate players who play 2 sports only. We can plug in the value of x = 2 in all the double overlaps in the diagram, i.e.: (7-2)+(5-2)+(4-2) = 5+3+2 = 10

So OA: 10.

Can one of the experts please tell us if 10 is correct or did we miss something.

Tanni, (with all due respect) could you please help us understand if we are making a mistake.

Thanks...

___________________
Karan

Hi MrR,
Just saw your video as well. First things first, that was quite thorough!!

But ya, as I said earlier I agree.

Thank you.

________
Karan

Hi All,
I still wanted to find out what was wrong with the formula given my Tanni, so was reading one of the replies(by ma127):

Gf + Gh + Gc - Gfh - Gfc - Ghc - 2(Gfhc) + None = Total
20 + 15 + 11 - Gfh - Gfc - Ghc - 2(Gfhc) + 18 = 50
64 - Gfh - Gfc - Ghc - 2(Gfhc) = 50
64 - (5-(Gfhc)) - (4-(Gfhc)) - (7-(Gfhc)) - 2(Gfhc) = 50
64 - 5 + (Gfhc) - 4 + (Gfhc) - 7 + (Gfhc) - 2(Gfhc) = 50
48 + (Gfhc) = 50
(Gfhc) = 2 ---> all three

If you read this solution carefully and compare it to the original formula given by Tanni:
Gf + Gh + Gc - Gfh - Gfc - Ghc - 2(Gfhc) + None = Total

You will see that what this formula essentially does is exactly what MrR explained. Read below for details.

When we start expanding the formula:

> Gf + Gh + Gc -(Gfh-Gfhc) -(Gfc-Gfhc) -(Ghc-Gfhc) - 2(Gfhc) + None = Total
> Gf + Gh + Gc - Gfh + Gfhc - Gfc + Gfhc - Ghc + Gfhc - 2Gfhc + None = Total [3 (+Gfhc) and 2 (-Gfhc) leave behind only 1 (+Gfhc)]
In conclusion,
> Gf + Gh + Gc - Gfh - Gfc - Ghc + Gfhc + None = Total [This is exactly what MrR and ma127 concluded]

I think this should conclude the issue and OA: 10

In case you get 13 or any other number as the answer then it is because you are applying the formula wrong. If I apply it wrongly then I get Gfhc as -1, see below:

>H+C+F-HC-FC-CF-2(FHC)+NONE = 50
>20+15+11-7-5-4-2(FHC)+18 = 50
>46-16-2FHC+18 = 50
>64-16-50 = 2FHC
>64-66 = -2 = 2FHC
> -1 = FHC (Wrong Answer)

So the formula given by Tanni is not wrong but its application/understanding may be wrong.

Thank you everyone.

_____________
Karan

The error in the formula I was using is in subtracting those elements that are counted three times. In fact, you have already subtracted those elements three times when accounting for the overlap for each pair of categories. Therefore the elements that appear three times have been eliminated three times and have to be added back in. You can find both versions online, but I now understand that you add in those counted three times rather than subtracting them again. so the correct formula is

Gf + Gh + Gc - Gfh - Gfc - Ghc +(Gfhc) + None = Total

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 does not play any of these given sports, ho many students play exactly two of these sports?

1. Draw a Venn Diagram
2. Plug in a value for the number of students who play all 3 sports.
3. Determine the other values in the Venn Diagram, working from the center out.
3. Check whether the sum of all the values = 50 students.

Since only 4 students play Cricket and Football, the number who play all 3 sports is likely to be 3 or less.
Let the number of students who play all 3 sports = 2.

Here's the Venn Diagram:



If 2 students play all 3 sports:
Since 7 in total play Hockey and Cricket, 7-2 = 5 play only Hockey and Cricket but not Football.
Since 4 in total play Cricket and Football, 4-2 = 2 play only Cricket and Football but not Hockey.
Since 5 in total play Hockey and Football, 5-2 = 3 play only Hockey And Football but not Cricket.

Working our way from the center out, we can determine the number who play exactly 1 sport:
Number who play only Hockey = 20-5-2-3 = 10.
Number who play only Cricket = 15-5-2-2 = 6.
Number who play only Football = 11-3-2-2 = 4.

Adding all the values in the circles to the 18 students who do not play any sports, we get:
(10+3+2+5+6+2+4) + 18 = 50.
Success!

The number who play exactly 2 sports = 5+2+3 = 10.