Three interviewers, A, B, and C are interviewing 40

Question

Three interviewers, A, B, and C are interviewing 40 applicants. Only with three interviewers' admission can an applicant be admitted. If interviewer A admitted 15 applicants, B admitted 17 applicants, and C admitted 20 applicants, at least how many applicants get the admission?

(A) 0
(B) 2
(C) 6
(D) 8
(E) 12

(A) 0
(B) 2
(C) 6
(D) 8
(E) 12

KarishmaB · Accepted Answer

There isn't much you need to do here for a graphical representation. Think of having a rectangle with 40 elements in it. Now draw a circle in it with 15 elements. Draw another circle which doesn't overlap this circle at all and has 20 elements in it. You have accounted for 35 elements and still have 5 elements leftover in the rectangle. Now the third circle with 17 elements can be drawn in any way inside the rectangle. There will be no elements lying in all three circles since the first two circles have no overlap. Hence it is not necessary that even one person gets admitted. Video for max min in Sets: https://youtu.be/oLKbIyb1ZrI

JeffTargetTestPrep · Answer

We can assume all 15 applicants admitted by A are also 15 of the 17 applicants admitted by B. Thus, so far, 17 applicants are admitted by at least one of the two interviewers A and B. However, since 17 + 20 = 37 < 40, it’s possible that none of these 17 applicants who are admitted by A and/or B are admitted by C, so the least number of applicants admitted by all three interviewers is 0.

Answer: A

timeishere · Answer

Answer is A = 0

If A admitted 15 are overlapping with B admission of 17 But C doesnot overlap with anybody.
Then no student will get nod from all the 3.
Hence 0 student will get admission.

jamifahad · Answer

i thought it was straight 12.
applicant interview by A + applicant interview by B + applicant interview by C - total.
15+17+20-40=52-40=12

Did i miss something?

timeishere · Answer

Isnt the ques saying that applicant has to get admission from all the three to get admission.
"Only with three interviewers' admission can an applicant be admitted."

pairakesh10 · Answer

Can anyone please post the graphical explanation as a solution for this problem ?

Divyadisha · Answer

'A' is the answer.

Question asks the LEAST number of applicants admitted. And, an applicant only be admitted when all three interviewers gave admission.

Interviewer C admitted- 20
Interviewer B admitted- 17
Interviewer A admitted- 15

Suppose 'B' admitted 17 candidates other than 'C' admitted. And then 'A' admitted remaining '3' (40-20-17=3) and any '12' from either 'C' or 'B'. In this case not even a single candidate is selected by all three, making the answer to be 'A'

BrentGMATPrepNow · Answer

Let's solve the question by completing a diagram with three overlapping circles.

There are 40 applicants in total 
15 + 17 + 20 = 52 
52 - 40 =  12 
So, to minimize the number of applicants approved by all three interviewers, we need to place  12  applicants in areas where two circles overlap.

Here's one possible scenario: 
 https://i.imgur.com/XIkBNX9.png

Since it's possible to have a scenario in which 0 applicants gain admission (i.e., 0 applicants are in the intersection of all three circles), the correct answer is A

egmat · Answer

GIVEN:    
  Total number of applicants = 40  
 Interviewers A, B, and C selected 15, 17, and 20 applicants, respectively.  
 An applicant gets admitted only when he is selected by  ALL three interviewers .

TO FIND:    
   Minimum number  of applicants to get admission.
  That is, the  minimum number  of applicants who are selected by  all  three interviewers.

SOLUTION:    
We will solve this question in some neatly laid out steps. Just follow each section and make sure you understand every bit of the solution. Let’s go!

Define variables:  
Let’s assign some variables first that will denote the various possibilities of selection: 
  Selected by  ALL three : 
  Say w = number of applicants selected by  all three interviewers    
 Selected by  EXACTLY two : 
  x = the number of applicants selected by  only A and B . 
 Y = the number of applicants selected by  only A and C . 
 z = the number of applicants selected by  only B and C .   
 Selected by  NONE : 
  N = the number of applicants selected by  none of the three interviewers .   
Note: Since nothing has been said about each applicant being selected by  at least one interviewer , there is a possibility that some of the applicants are selected by  none of the interviewers .

Represent on Venn diagram:  
Now, we’ll draw a Venn diagram that will clearly show the scenario at play here. We will use the variables as we defined above. Here goes:
 https://i.imgur.com/iRlSb4N.png

Despite populating all we know, there still are some empty regions in our Venn diagram. But it’s no big deal. Using what the diagram already has, we can easily find the values of the empty regions as well. It’s always a good idea to understand all possible regions! 
So, here’s what we can infer: 
  Number of applicants selected by  only A  = 15 – x – y – w. (This is the part of circle A that does not meet any other circle.)  
 Number of applicants selected by  only B  = 17 – x – z – w. (This is the part of circle A that does not meet any other circle.)  
 Number of applicants selected by  only C  = 20 – y – z – w. (This is the part of circle A that does not meet any other circle.)

And done! Now, it’s time for the main question asked – the minimum possible value of ‘w’.

Finding Minimum possible ‘w’:  
Looking at the Venn diagram, we can say that: 
 Total applicants  = Selected by  only A  + Selected by  only B  + Selected by  only C  + Selected by  only A and B  + Selected by  only A and C  + Selected by  only B and C  + Selected by  all A, B and C  + Selected by  None  ----(I)

Rewriting the equation using all our variables, we get: 
  40 = (15 – x – y – w) + (17 – x – z – w) + (20 – y – z – w) + x + y + z + w + N 
⇒ 40 = 52 – 2w – x – y – z + N 
⇒ 2w = 12 - (x + y + z) + N ----(II)

Observe that to minimize ‘w’, we need to  maximize (x + y + z) and minimize (N) . Why? 
 - Because the larger a value we subtract from 12, the smaller the remaining difference will be, leading to a smaller ‘w’. 
- And the smaller a value we add to 12, the smaller the resulting sum will be, leading to a smaller ‘w’.

PART 1: Minimize N  
At the least, N can take a minimum value of  ZERO . This is when each of the 40 applicants were selected by at least one of the three interviewers. 
In this case, (II) becomes: 
 - 2w = 12 - (x + y + z) ----(III)

PART 2: Maximize (x + y + z)  
Note that ‘w’, being a certain number of applicants, cannot be negative. Hence, the maximum that we can subtract from 12 is 12 itself. That is maximum (x + y + z) = 12.
Hence, from (III), we get 2w = 12 – 12 = 0.

This implies that the  minimum possible  value for w is  ZERO .

Correct Answer:   Option A

Hope this helps!

Best, 
Aditi Gupta 
Quant expert, e-GMAT

chetan2u · Answer

ChandlerBong

We are looking for the worst scenario where none or the least of the person are selected by all three.

More than maths, it is logic based. 
For maths, you can make three circles with overlap and fit the numbers to ensure minimum in the area overlapped by all three circles.

Pure logic
Let the applicants be named 1 to 40.
A: Selects 1 to 15
B: Doesn’t select 1 to 15, but 16 to 32.
Now, there is no overlap between A and B, so surely irrespective of whom C selects, there will be none who would have been selected by all three.

So 0 is the answer.

GGGMAT2 · Answer

Thanks for the explanation, really helpful.

If the question would have been - Three interviewers, A, B, and C are interviewing 40 applicants. Only with three interviewers' admission can an applicant be admitted. If interviewer A admitted 15 applicants, B admitted 17 applicants, and C admitted 20 applicants, at most how many applicants get the admission?

Then the answer would be 15 right?

KarishmaB

KarishmaB · Answer

Yes, this point is also explained in detail with the Venn digram in the video of max min: https://youtu.be/oLKbIyb1ZrI

HarshavardhanR · Answer

https://gmatclub.com/forum/download/file.php?id=83581 The visual above shows one possible scenario. Harsha GMAT-Club-Forum-dmf5stqi.png

sequimaxime · Answer

A three overlapping set question asking for min overlap 
a + b + c - 2N 
15+17+20-80 =-28 
so max(-28,0)=0 
hence A

Usernamevisible · Answer

This is an overlapping sets minimum intersection question.
We need the minimum number admitted by all three.
Let:
A admits 15
B admits 17
C admits 20
Total applicants = 40
Total admissions given out:
15 + 17 + 20 = 52
If nobody were admitted by all three, how many admission "slots" can 40 applicants absorb?
To minimize triple overlap, spread the 52 admissions as much as possible.
Each applicant can receive at most 2 admissions without becoming a triple-overlap applicant.
So 40 applicants can absorb at most:
40 × 2 = 80
admissions.
Since only 52 admissions exist, there is plenty of room to distribute them with no applicant receiving all three approvals.
For example:
• 15 applicants approved by A and B
• 2 applicants approved by B and C
• 3 applicants approved by A and C
• Remaining approvals can be distributed among single-approval applicants
The exact arrangement can be adjusted, but the key point is that 52 approvals can be distributed among 40 people without forcing any triple overlap.
----------------------------------------------------------------------------------------------------------------------------------------------------

A more formal test:
Minimum triple overlap is forced only if:
A + B + C > 2N
because after every applicant has received 2 approvals, any extra approvals must create triple overlaps.
Here:
52 < 2(40) = 80
so no triple overlap is required.
Therefore the minimum number admitted by all three is:
0
Answer: (A) 0
-------------

GMAT Takeaway
For three-set minimum intersection problems:
• Compute total memberships S = A + B + C
• Compare with 2N
• Minimum triple intersection = max(0, S − 2N)
Here:
max(0, 52 − 80) = 0
So the minimum number admitted by all three interviewers is 0.

Three interviewers, A, B, and C are interviewing 40

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Three interviewers, A, B, and C are interviewing 40

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