T=n(A)+n(B)+n(C)-n(Exactly two of the events)-2*n(All 3 Events)+n(None of the events)

T=144
n(A)=T-n(A')=144-89=55
n(B)=T-n(B')=144-100=44
n(C)=T-n(C')=144-91=53
n(Exactly two of the events)=n(At least 2 Events)-n(All 3 Events)=37-6=31
n(All 3 Events)=6

144=55+44+53-31-2*6+n(None of the events)
n(None of the events)=144-55-44-53+31+12=35

Ans: "D"

people who could swim =55

People who clould dance =44

people who could drive = 53

total number of driving,swiming and dancing events = 55 +44 +53= 152

6 people have been triple counted for their expertise, same people extra counted 12 times.

(37 - 6) = 31 people have been double counted for their expertise.

total number of different people who could do these different activities = 152 -(12 +31) =109

so people who could do nothing 144 -109 = 35

Answer D

people who could swim =55

People who clould dance =44

people who could drive = 53

total number of driving,swiming and dancing events = 55 +44 +53= 152

6 people have been triple counted for their expertise, same people extra counted 12 times.

(37 - 6) = 31 people have been double counted for their expertise.

total number of different people who could do these different activities = 152 -(12 +31) =109

so people who could do nothing 144 -109 = 35

Answer D

please could you elaborate as to how did you get the individual nos

I used to take more time to solve this type of question. Fluke's method is awesome and will help to reduce the time to solve.
Thanks Fluke!!!!!!!!

You're welcome!!! The appreciation goes to the contributors of this thread:

formulae-for-3-overlapping-sets-69014.html
esp.
formulae-for-3-overlapping-sets-69014.html#p729340

#not swim = 89 => #swim = 144-89 = 55
#not dance = 100 => #dance = 144 - 100 = 44
#not drive = 91 => #drive = 144 - 91 = 53

at least two = sum of two's + sum of three's = 37
sum of three's = 6
=> sum of two's = 31

total = S + D + Dr - (sum of two's ) - 2*(sum of three's) + none

144 = 55 + 44 + 53 -31 - 2*6 + none

=> none = 35

Answer is D.

+1 for D.

Use 3 set formula:

Total = Group1 + Group2 + Group3 - (sum of 2-group overlaps) - 2*(all three) + Neither

Found individual numbers as explained in the earlier posts.

Hey liftoff,

I used to be confused with such kind of problems and what helped me is drawing Venn. If you follow this there is no need to remember any formula. One just need ot remember simple +/- and also take less than 2 min.

Cheers

I am not sure that stated official answer is correct one. or I am tricked by the language (which generally is the case in these kind of question). IMHO, the number of people who could do at least two of the things also includes 18 ( 3 *6 where 6 are the number of people who could all these things). My answer is coming as option B - 23. Can anybody help.

The number of people who can do at least two things includes 6 (number of people who can do all three), not 6*3.

Understand here that 37 is the number of people, not the number of instances. Hence 6 is not counted 3 times in 37. Out of 37 people, 18 people cannot do all three. Only 6 can do all three. So 31 can do exactly 2 things.

I have discussed this concept in my blog post given below:
http://www.veritasprep.com/blog/2012/09 ... ping-sets/

Question 1 on the diagram is this.
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Hello All,

I started as Da, Dr, Sw, be the number of people who could only dance, Drive and swim respectively

X is number of people who could dance and drive
Y is number of people who could dance and swim
Z is number of people who could drive and swim.

so

X+ y+ z=37-6=31

Da + Dr + X = 89
Da + Sw + y = 91
Dr + Sw + z = 100

2(Da + Dr + Sw) +x+y+z = 89+91+100=280
2 (Da + Dr + Sw) + 31 = 280
2 (Da + Dr + Sw) = 280-31 = 249

where did i go wrong here???

Not correct. 89 includes the people who cannot do any of the three things. So does 91 and 100.
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But the stem says "at least 2", not exactly. so it should be the formula as follows:

T=n(A)+n(B)+n(C)-n(at least two of the events)+n(All 3 Events)+n(None of the events)

so 144=55+44+53-37+6+X

144=152-37+6+X
144=121+X
X=23

OA is incorrect,

How do you know that's true though; to me the question read like every other overlapping data set question. It didn't specify 'exactly' two, and seemed to be worded as other questions which mention people being in at least two sets...

The statement given in the question is this:
"If the number of people who could do at least two of these things, was found to be 37 and the number of people who could do all these things was found to be 6"

Say, I have 37 people in front of me and I say that these are the people who can do at least two of the three things - say these people are P1, P2, ...P37.
I also know that exactly 6 people can do all three things. These 6 are P1, P4, P8, P9, P10, P12
Tell me, how many people can do exactly 2 of the three things? 31 or 19? The answer here is 31.

Note that this situation is different from the usual: 10 people can swim and dance, 20 people can dance and drive and 7 people can swim and drive. In this case, each of the 10, 20 and 7 includes the people who can do all 3 things and hence 10 + 20 + 7 - 6*3 = 19 people can do exactly two things.
_________________

Using the formula is a bad idea if you don't understand exactly when and how to use it. If you understand exactly when and how to use the formula, then you would find it too cumbersome to use it and will anyway prefer to reason out the answer!

In your formula, n(at least two of the events) is the sum of the intersection of the circles. This means each intersection includes the area where only two overlap and where all 3 overlap.
To check out the two formulas, check out this link: a-school-has-3-classes-math-class-has-14-students-150221.html#p1207266
After you check out the link, note that in your formula, n(at least two of the events) = (d + g) + (e + g) + (f + g)
whereas the 37 given to you in this question is (d + e + f + g)
_________________

We see that 144 - 89 = 55 residents could swim, 144 - 100 = 44 could dance and 144 - 91 = 53 could drive a car. We can use the formula:

Total = #(swim) + #(dance) + #(drive) - #(exactly two groups) - 2 * #(all three groups) + #(neither)

We need to find #(neither), so let’s denote it by n. We have the numbers for all the other components of the formula except for #(exactly two groups). That is because the number 37 represents #(at least two groups), in other words, 37 represents #(exactly two groups) + #(all three groups). So:

37 = #(exactly two groups) + 6

31 = #(exactly two groups)

Now, we can use the formula:

144 = 55 + 44 + 53 - 31 - 2*6 + n

144 = 109 + n

35 = n

Answer: D
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HiVeritasKarishma,

Can you explain me " Out of 37 people, 18 people cannot do all three. Only 6 can do all three. So 31 can do exactly 2 things."
How did u get 18 18 people cannot do all three.
Well my understanding is if 6 people can do all three events and 37 can do at least two events , then 31 are the ones that can do exactly two events. But how did u derive that 18 people cannot do all three.

All we know from total of 144 : exactly two events 31, exactly three events is 6, none of the events is 35, only single events 72.

Probus
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Request someone explain to me how " 89 could not swim" translates to
144-89 could swim