I12-21

Question

A candidate applying for a research program must undergo up to three interview cycles. The first interview is conducted by A, followed by B, and, if necessary, by C. A candidate is selected as soon as exactly two interviewers provide a recommendation. For example, if both A and B recommend a candidate, the candidate is selected and does not proceed to an interview with C. C interviews only those candidates recommended by exactly one of A or B.
 
A total of 132 candidates applied for the research program, of which C interviewed 82. The number of candidates recommended by A, B, and C were 70, 86, and 50, respectively.
 
Based on the information provided, select for  Both A and B  the number of candidates recommended by both A and B, and select for  Rejected  the total number of candidates out of 132 who were not selected for the research program.

Bunuel · Answer

Official Solution:

Denoting the candidates recommended by both A and B as "Both," the number of candidates recommended by only A is 70 - Both, and the number of candidates recommended by only B is 86 - Both. Since C interviewed 82 candidates, and C only interviews those recommended by exactly one of A or B, we have (70 - Both) + (86 - Both) = 82, which gives  Both = 37 .
 
Therefore, both A and B recommended 37 candidates, and since C (together with A or B) recommended 50 candidates, the total number of candidates selected for the program is 37 + 50 = 87. Thus,  the number of candidates not selected for the program is 132 - 87 = 45 .

Correct answer:

Both A and B  "37"

Rejected  "45"

Bunuel · Answer

I have revised the question, solution, and formatting by adding more details to enhance clarity.

dnguyen1203 · Answer

I think this is a  high-quality  question and I agree with explanation.

sindhureddyalla · Answer

I like the solution - it’s helpful.

Nipunh · Answer

another way to approach this question is by writing down all the possible combinations, and just noting values, on simplifying we will get column 1&2 as 50-a, 32+a format, based on which we can mark answer and continue.

adarsh2209 · Answer

I like the solution - it’s helpful.

Karym · Answer

I like the solution - it’s helpful.

BongBideshini · Answer

I like the solution - it’s helpful.

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