64 candidates are competing for 5 positions at a consulting

64 candidates are competing for 5 positions at a consulting firm. The hiring process consists of 3 interviews. After each interview, n% of the remaining candidates will be dismissed. The candidates will be selected from among those complete all three rounds. Each candidate is equally qualified and has an equal probability of getting hired at every point in the process. What is the probability that a candidate will complete all three interviews but fail to get the job?

(1) n = 25
So \(75%\) remains after each interview
\(64=>48=>36=>27\), \(27\) will arrive at the last step. So the probability that a candidate will complete all three interviews but fail to get the job is \(\frac{22}{64}\).
Sufficient

(2) 12 candidates completed the first interview but were dismissed after the second interview.
So there is a difference between the first and second interview of 12. People who completed the first interview = \(64(\frac{100-n}{100})\); people who completed the second interview = \(64(\frac{100-n}{100})^2\).
Call \(x=\frac{100-n}{100}\) and solve the equation \(64x=64x^2+12\) \(x=0.75\) => \(n=25%\) or \(x=0.25\) => \(n=75%\).
If \(n=75%\) this means that they keep 25% each time, so the sequence would be \(64=>16=>4=>1\); in this scenario there are no 5 people to be chosen for the final 5 spots, so \(n=75%\) is not a realistic rate (given the conditions above).
If \(n=25%\) the sequence is the same as A. So IMO sufficient

IMO D

Start - 64
1st Interview : 64x (where x= 1-n/100)
2nd Interview: 64x²
3rd Interview: 64x³
Getting a job: 5 (No relation to above.
What is asked?
(65x³-5)/64

Statement 1:
n=25
So, x=0.75
So, probability = 64*3/4*3/4*3/4 or (27-5)/64 = 22/64
SUFFICIENT

Statement 2:
Given: 64x-64x² = 12
Solve this quadratic equation: 16x²-16x+3=0
You get x=1/4 or x=3/4
BUT
If x=1/4, 64x² becomes 4. As no. of students who got the job is 5, so no. of students left after 2nd interview can't be less than this number.
Check 3/4, you get the case as in statement 1., so same answer.
SUFFICIENT

EITHER SUFFICIENT

All you need to answer the question is the value of n and each statement provides sufficient information to get it, thus the answer is D.
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people who completed the second interview = \(64(\frac{100-n}{100})^2\).




how did we derive this , kindly show me the light

People who survive the first interview is \(64(\frac{100-n}{100})\).

Then \((\frac{100-n}{100})\) of that number survive the second one.

Unite those equation into \(64(\frac{100-n}{100})(\frac{100-n}{100})=64(\frac{100-n}{100})^2\)

Numbers can help here. If n=25% then 75% survive each step

\(64*0.75=48\) I interview
\(48*0.75=36\) II interview

\(64*0.75*0.75=64*(0.75^2)=36\) II interview.

Is it clear? let me know

aah ! I get it now thx

Its mentioned n% of the remaining candidates.
So how are we taking 64 into consideration.
64 is not the 'remaining candidates' its total 'initial candidates'.

Have I read the statement II incorrect ?
lets consider, n - percent of people not selected
I think statement II instead meant,
64(100-n)/100 *(n/100)=12

Please share your thoughts

can someone please help me find a good way to factor (2) to get 1/4 and 3/4? I know the conventional way, but the fraction part is proving challenging to identify

The second statement says that 12 candidates made it after the first interview and were disqualified.
This however doesnt talk about the other candidates who qualified with them after the first interview and survived the second interview as well.
Therefore, insufficient.

statement 2 is possible in two cases

n=12.5%

n=50%

Hence A

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