When we've got three groups A, B, and C with some overlap, we can say that

Total = A + B + C + (people in no groups) - (people in exactly two groups) - 2*(people in exactly three groups)

We're told that the total = 144, A = (144 - 100), B = (144 - 89), and C = (144 - 91). (Remember that A, B, C are the people who CAN do each of the three activities, and Can = Total - Can't.)

We're also know that (people in exactly three groups) = (people in at least two groups) - (people in exactly two groups), so people in exactly three groups = 6 and people in exactly two groups = 31.

From there, our equation is

144 = (144 - 100) + (144 - 89) + (144 - 91) + (people in no groups) - 31 - 2*6

which, after some tedious arithmetic, gets us

35 = people in no groups

swerve wrote:A survey was conducted to find out how many people in a housing colony of 144 residents could swim, dance a drive a car. It was found that the number of people who could not swim was 89, the number of people who could not dance was 100 and that the number of people who could not drive a car was 91. 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, how many people could not do any of these things?

A) 17
B) 23
C) 29
D) 35
E) 50

The OA is D.

Please, can any expert explain this PS question for me? I have many difficulties to understand why that is the correct answer. Thanks.

let us take number of people who could only swim = a
...........................................................................................drive = b
...........................................................................................dance = c
...........................................................................................swim and drive = d
............................................................................................drive and dance= e
...........................................................................................dance and swim = f
..................................................................................not do any of the items= Z
number of people who know swiming+driving+dancing = 6
......................................................... at least two items = 37

therefore a+b+c+d+e+f+Z+6= 144

and d+e+f = 37 - 6 = 31

therefore a+b+c+ 31 +Z+6= 144

or a+b+c = 107- Z

those who could not swim = 89
therefore b+c+e+ Z = 89
Similaly a+c+f+ Z = 91
and a+b+d+ Z = 100
Adding we get 2(a+b+c) + (d+e+f) + 3Z =280
or 2(107-Z) + 31 + 3Z = 280

This gives Z = 280 - 214-31 = 35

Hence D is the correct answer

swerve wrote:A survey was conducted to find out how many people in a housing colony of 144 residents could swim, dance a drive a car. It was found that the number of people who could not swim was 89, the number of people who could not dance was 100 and that the number of people who could not drive a car was 91. 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, how many people could not do any of these things?

A) 17
B) 23
C) 29
D) 35
E) 50

The OA is D.

Please, can any expert explain this PS question for me? I have many difficulties to understand why that is the correct answer. Thanks.

Got a message to provide an alternative method to solve the problem.

So here it is, using set theory.

Let the set of people who can swim be A, the set of people who can dance be B and the set of people who can drive be C. The total number of people i.e. Universe set (U) in the figure below is 144.



We have to find (A∪B∪C)^c where (A∪B∪C)^c is the complement of (A∪B∪C)
(A∪B∪C)^c = U- (A∪B∪C) = 144 - (A∪B∪C) ...................... equation 1.
We know that A^c = 89 or 144 - A = 89. i.e. A = 144 - 89 = 55. Similarly, B = 44 and C = 43 ...................equation 2.

In set theory, number of people in two or more sets is (A ∩ B) + (B ∩ C) + (C ∩ A) -2 (A ∩ B ∩ C) and we know that this number is 37.

We also know that (A ∩ B ∩ C) = 6 ...................equation 3.

Hence (A ∩ B) + (B ∩ C) + (C ∩ A) = 37 + 2 * 6 = 49...................equation 4.

In set theory, (A∪B∪C)=A + B + C - (A ∩ B) - (B ∩ C) - (C ∩ A) + (A ∩ B ∩ C)
Putting all the values from equation 2, 3 and 4 the above equation, (A∪B∪C) = 55 + 44+ 43 - 49 + 6 = 109 ...................equation 5.
Using equation 1 and 5
(A∪B∪C)^c = 144 - 109 = 35.

Hence D.

swerve wrote:A survey was conducted to find out how many people in a housing colony of 144 residents could swim, dance a drive a car. It was found that the number of people who could not swim was 89, the number of people who could not dance was 100 and that the number of people who could not drive a car was 91. 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, how many people could not do any of these things?

A) 17
B) 23
C) 29
D) 35
E) 50

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