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# To get a job at Company X, an applicant must be recommended by

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Re: To get a job at Company X, an applicant must be recommended by [#permalink]
15+17+20 =52 -30 = 22 that could overlap , how do we know we have to allocate 2 to 15 and 17 and the rest to third category ?
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Re: To get a job at Company X, an applicant must be recommended by [#permalink]
diya7 wrote:
15+17+20 =52 -30 = 22 that could overlap , how do we know we have to allocate 2 to 15 and 17 and the rest to third category ?

22 is the maximum number of applicants who could have been recommended by more than one interviewer. The question, however, asks for the least number of applicants recommended by all three interviewers. For that, we first find the least number that both the first and second interviewer could have recommended, which is 15 + 17 - 30 = 2. Then, we check if the 20 applicants recommended by the third interviewer can be distributed such that they don't overlap with these 2. Since there are 30 applicants in total, this is possible: the 2 applicants recommended by both the first and second interviewers could have been overlooked by the third interviewer, who could have recommended any 20 out of the remaining 28 applicants.
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Re: To get a job at Company X, an applicant must be recommended by [#permalink]
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JessicaJ wrote:
To get a job at Company X, an applicant must be recommended by three interviewers. Out of 30 applicants, 15 were recommended by the first interviewer, 17 by the second and 20 by the third interviewer. What is the least number of applicants who would have to have been recommended by all three interviewers?

A. 0
B. 2
C. 3
D. 5
E. 10

­Max min in overlapping sets is discussed here: https://youtu.be/oLKbIyb1ZrI

Start with the largest set and put it down: 20. We have 10 remaining.
When we put down 17, we avoid overlap so 10 go to the remaining and 7 overlap with the 20 set.
When we put down 15, to avoid overlap, we put all 15 (shaded region) on the 23 where there is no overlap. Hence minimum overlap of all three is 0.

Attachment:

Screenshot 2024-06-24 at 2.35.11 PM.png [ 53.28 KiB | Viewed 1189 times ]

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Re: To get a job at Company X, an applicant must be recommended by [#permalink]
KarishmaB wrote:
JessicaJ wrote:
To get a job at Company X, an applicant must be recommended by three interviewers. Out of 30 applicants, 15 were recommended by the first interviewer, 17 by the second and 20 by the third interviewer. What is the least number of applicants who would have to have been recommended by all three interviewers?

A. 0
B. 2
C. 3
D. 5
E. 10

­Max min in overlapping sets is discussed here: https://youtu.be/oLKbIyb1ZrI

Start with the largest set and put it down: 20. We have 10 remaining.
When we put down 17, we avoid overlap so 10 go to the remaining and 7 overlap with the 20 set.
When we put down 15, to avoid overlap, we put all 15 (shaded region) on the 23 where there is no overlap. Hence minimum overlap of all three is 0.

Attachment:
Screenshot 2024-06-24 at 2.35.11 PM.png