A survey was conducted to determine the popularity of 3 food

\(Total = A + B + C - (sum \ of \ 2-group \ overlaps) + (all \ three) + Neither\).

75 = 48 + 45 + 58 - (28 + 37 + 40) + 25 + Neither --> Neither = 4.

Only Pizza = P - (P and H + P and T - All 3) = 48 - (28 + 40 - 25) = 5;
Only Hoagies = H - (P and H + H and T - All 3) = 45 - (28 + 37 - 25) = 5;
Only Tacos = T - (P and T + H and T - All 3) = 58 - (40 + 37 - 25) = 6.

The number of students who like none or only one of the foods = 4 + (5 + 5 + 6) = 20.

Answer: D.

For more check ADVANCED OVERLAPPING SETS PROBLEMS: advanced-overlapping-sets-problems-144260.html

Hope this helps.
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I have edited the question. The total number of students was missing. After that I could solve it as well. The updated solution is attached.

Make a Venn diagram. The toughest of sets questions can be done easily through venn diagrams because you can visualize easily.

Draw three overlapping circles (as shown by Bunuel in his post above). Mark set A as Pizzas, set B as Hoagies and set C as Tacos.
The part where all three overlap (g in the diagram), mark that as 25.
28 like pizza and hoagies. 25 like all three so 3 like only pizzas and hoagies. Mark d as 3.
37 like hoagies and tacos. 25 like all three so 12 like only hoagies and tacos. Mark f as 12.
40 like pizza and tacos. 25 like all three so 15 like only pizza and tacos. Mark e as 15.

From 75, subtract d, e, f and g. This will give the number of students lying in a, b, c and None. This is exactly what we want.
75 - 3 - 12 - 15 - 25 = 20
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Bunuel / VeritasPrepKarishma, is there a simpler method similar to the table method used for such problems? This appeared in the first 10 questions of my test and I was stumped by the numbers involved.

This one is tricky and lengthy , I guess it doesnt end there in Knowing your Formula 1 and Formula 2 :D ..... I tried solving this and then I got stuck again solving for exactly 1 like . Though it was preety straight forward for solving Neither .

Also I recognize something , for the same problem I can also number of people who like exactly 2 like

using

Total = A+B+C - ( Exactly 2 like AB+BC+CA) - 2 ABC +Neither.

Hence

75 = 151 - Exactly 2 like - 2 (25) + 4

Exactly 2 like =26 .

Correct me if I am missing something .I guess though its the right approach . Also for this we need to primarily work 1st step Using Formula 1 to find Neither . Then only its possible . Suggest me if there is any other approach
Thanks.

How are we assuming about the neither part to be . I mean when we are subtracting (All three) from each of the (2 like region )

we assume that its part of the bigger a , b and c .

I mean suppose we try solving for Exactly two like using the approach above then we have ,

All 2 like = (28-25)= 3 ; (37-25)=12 ; (40-25)=15 ; Therefore 3+12+15=30 .....But I guess its wrong as I am missing out on the Neither Part.

Now If I consider the approach used by Bruneal then I know that Neither =4 ; so Exactly liking 2 items would be 30-4=26 .

Please correct me if I am wrong , also what approach should I take for Neither.

The question asks you the sum of 'None' and 'Only 1' i.e. it asks you 'None + Only 1' (irrespective of how many are in None and how many are in Only 1). You just need the sum.

You have that the total number of students = 75

75 = None + Only 1 + Only 2 + All 3
To get 'None + Only 1', all you need to do is subtract 'Only 2' and 'All 3' from 75.

'Only 2' =3 + 12 + 15 = 30 (as correctly calculated by you. It does not include 'None')
'All 3' = 25

So None + Only 1 = 75 - 30 - 25 = 20
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There's a faster way of doing it.
Let x= total with 3 foods
Total with with 2 foods only => (37-x) + (28-x) + (40-x) = 105-3x= 30
Total with 3 = x = 25

Total 0 and 1 = Total - (Total 2 and Total 3)
= 75 - 30 - 25
= 20

Dear Bunuel,
I think in your solution, you intended to use the second formula from the gmatclub mathbook,but wrote the first formula.
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Can you elaborate on this? Why is the formula used wrong?
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I'd appreciate further clarification on why we use the version of the formula where we add back in the 'all' category? I'm not sure why we are able to assume that the 'all' category also includes numbers from the 'two items' sections? Thanks

To understand the formulas please check ADVANCED OVERLAPPING SETS PROBLEMS

Hope this helps.
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When you draw the Venn diagram, I suggest starting with the "triple overlap" number of 25 and working your way out from there. Don't forget to adjust the numbers as you go.

Attached is a visual that should help.

The thing that's important to remember with this time of problem is that it says "37 like hoagies and tacos" not "37 like only hoagies and tacos" ---> knowing this we can apply the first of 2 formulas used for 3 overlapping sets

total= A + B + C - [sum of 2] + [all three] + neither
75= 71 + x
Neither= 4

Work in reverse to find the number of exactly 2

D

The question could be solved in two parts:-

1) First 75 minus the number contained in union of the three sets will give the number of students who like none of the foods.(i.e. 75-71=4)
2) In second part, number of students who like only one of the foods = (71-55=16)

Total = 4+16 = 20

Option D is the correct answer.
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CAMANISHPARMAR, how do you find 4 from first part (or 71)?

ScottTargetTestPrep

Hey Scott,

I was wondering why formula given in TTP formula sheet won't work: Unique members = All A + All B + All C - Groups of two - 2 group of three + Neither.

If I plug in I find that Neither is 71.

I solved all ttp questions and didn't get anything like this

75 = 48 + 45 + 58 - (28 +37 +40) - 2(25) + Neither

I also see this formula being cited:Total=A+B+C−(sum of 2−group overlaps)+(all three)+Neither

I haven't seen this formula on TTP either

Thank you

I understand that some of the quantities in the formula need clarification. For instance, in the formula:

Unique members = All A + All B + All C - Groups of two - 2 group of three + Neither

“Groups of two” refers to the number of members who belong to exactly two of the groups. For instance, in the question it says 28 people like pizza and hoagies; however, the 28 people also include people who like pizza, hoagies AND tacos. For the formula to work, you need to take the number of people who like ONLY pizza and hoagies to be 28 - 25 = 3. Similarly for the remaining groups of two.

If you take the number of unique members to be 75; All A, All B and All C to be 48, 45 and 58 respectively; groups of two to be 30 (because there are 28 - 25 = 3 who like pizza and hoagies only, 37 - 25 = 12 who like pizza and taco only, 40 - 25 = 15 who like pizza and taco only, for a total of 3 + 12 + 15 = 30) and groups of 3 to be 25; you’ll get:

75 = (48 + 45 + 58) - 30 - 2(25) + neither

75 = 151 - 30 - 50 + neither

neither = 4

By the way, if you solve 75 = 48 + 45 + 58 - (28 +37 +40) - 2(25) + Neither; you’ll get neither = 79, not 71.

If you’d rather use the formula Total=A+B+C−(sum of 2−group overlaps)+(all three)+Neither, then you need to take 48 + 45 + 58 = 105 for the quantity “sum of 2-group overlaps”.

Posted from my mobile device
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E = Exactly

Total = E1 + E2 + E3 + None
75 = (E1 + None) + E2 + 25

Find E2:
E2 = (28 - 25) + (37 - 25) + (40 - 25) = 3 + 12 + 15 = 30

E1 + None = 75 - E2 - 25 = 75 - 30 - 25 = 20