64 candidates are competing for 5 positions at a consulting

Question

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
(2) 12 candidates completed the first interview but were dismissed after the second interview.

(1) n = 25
(2) 12 candidates completed the first interview but were dismissed after the second interview.

Zarrolou · Accepted Answer

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
(2) 12 candidates completed the first interview but were dismissed after the second interview.

(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

mkdureja · Answer

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

Bunuel · Answer

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.

IvyLeague56 · Answer

people who completed the second interview = 64(\frac{100-n}{100})^2 . how did we derive this , kindly show me the light :shock:

Zarrolou · Answer

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

IvyLeague56 · Answer

aah ! I get it now

thx

Adit_ · Answer

Bunuel  Can I assume that all of them are taking part in the interviews each time?
It could also be the case that they are sent in batches since the question stem doesnt mention parallel conduction of all interview and n% of "remaining" gets dismissed so I was thinking we can't figure out unless we know how many take part per interview
Kindly clarify.

CuriousThinker · Answer

Can someone help me understand the equation in the 2nd statement

Nidhibatra · Answer

Wouldnt it take a lot of time to check if statement 2 has one possible solution or two? How to do that quickly?

egmat · Answer

Hi Nidhibatra,

You're worrying about a step that's actually a lot faster than it looks. You don't need to run both roots through the whole 3-round chain to see which one dies.  One check, on one number, does it.

The shortcut: test only the smaller root

Once the quadratic gives you x =  1/4  or x =  3/4 , ask a single question:  can the survival rate be x = 1/4?

The stem hands you a hidden floor:  5  candidates must finish all three rounds (they're the ones getting hired). Survivors after  3  rounds =  64  · x3. So test just the smaller root:

- x =  1/4  - survivors =  64  · ( 1/4 )3 =  64/64  =  1 . Only  1  candidate left - can't fill  5  seats.  Reject.

That's it. The larger root (x =  3/4 ) is the only one left, so it must be the answer - no need to compute anything for it. The statement is  sufficient  because exactly one root survives the real-world constraint.

Why you can skip the bigger root

Survival rate x =  3/4  keeps  more  people at every round than x =  1/4 . If the smaller x fails the " 5  must remain" floor, the bigger x can only do better - you never need to check it. This is why you always test the  smaller  root first: it's the one most likely to break the constraint, and killing it settles the question.

The habit to build

Whenever a DS quadratic gives two roots, don't re-solve the whole problem twice. Instead:

1. Find the  physical constraint  buried in the stem (here: " 5  positions must be filled," so ≥ 5  must reach round  3 ).
2. Apply it to the  smaller  root - one cube, one comparison.
3. If it fails, the other root wins automatically. Statement is  sufficient .

On this question, that's a  single line of arithmetic  -  64  · ( 1/4 )3 =  1  <  5  - and the second root vanishes. That's the "quick" way you were looking for.

Answer: D

egmat · Answer

Hi CuriousThinker,

The equation in Statement (2) is just tracking  head counts round by round , so let me rebuild it slowly from the posts by Zarrolou and mkdureja.

Let  x = (100 - n)/100  = the fraction of people  kept  after each interview.

- Start:  64  candidates.
- Survive interview  1 : 64x.
- Survive interview  2 : each survivor of round  1  again has fraction x retained, so 64x · x = 64x2.

Now read the statement carefully:  "12 candidates completed the first interview but were dismissed after the second."  Those are people who were  in  round  2  (they survived round  1 ) but did  not  survive round  2 . So:

dismissed after interview  2  = (survived interview  1 ) - (survived interview  2 )

64x - 64x2 =  12

That's exactly the equation. Rearranged it becomes 16x2 - 16x + 3 =  0 , giving x =  0.75  (n =  25 ) or x =  0.25  (n =  75 ).

A quick number check to feel it  - take n =  25 , so x =  0.75 :
- After round  1 :  64  ×  0.75  =  48 
- After round  2 :  48  ×  0.75  =  36 
- Dismissed between those two rounds:  48  -  36  =  12  ✓

That's the whole idea: "dismissed after round  2 " is simply the drop from the round- 1  count to the round- 2  count, which is 64x - 64x2.

One last piece so the statement stays  sufficient : n =  75  would leave only  1  person after three rounds ( 64  -  16  -  4  -  1 ), too few to fill  5  seats - so it's rejected, leaving n =  25  as the only valid value. That pins the probability down, which is why ( 2 ) alone works.

Answer: D

PradhyumnanD · Answer

There is no constraint that the 5 openings must be filled. The question is clear: there are five openings. It has not been mentioned that all the positions were filled. In that case A would be the right answer.

egmat · Answer

Hi PradhyumnanD,

You've put your finger on the  exact  pivot of this whole question, so let's settle it head-on.

You're right that the stem never writes the words "all 5 positions are filled." But look at what it  does  say: "64 candidates are competing for  5 positions " and "the candidates will be  selected from among those who complete all three rounds ." That's not just describing empty slots - it's telling you that  5 people are chosen  from the finishers. For  5  to be selected, at least  5  people have to survive all three interviews.

Why n = 75 breaks that

Run the smaller-survival case the way the thread did:

- n =  75  keeps only  25%  each round:  64  →  16  →  4  →  1 .
- That leaves exactly  1  finisher.

You cannot select  5  candidates from a pool of  1 . So n =  75  doesn't just look unrealistic - it directly  contradicts a fact the stem gives you  (that  5  are selected). It's an invalid solution to the equation, the same way a "length =  -3 " answer gets thrown out of a geometry problem.

Why this matters for sufficiency

Here's the part that shows why it can't be A. If you  did  allow n =  75 , the two roots give genuinely different answers:

- n =  25  →  27  finishers → probability =  22/64 
- n =  75  →  1  finisher,  5  can't be picked → the setup falls apart

Only n =  25  is consistent with the premise, so Statement ( 2 ) pins down a  single  valid value of n - which is all the question needs. That's why it's  sufficient , giving  D .

The takeaway: a stated quantity like " 5  positions filled" is a real constraint, not decoration - use it to discard roots that violate it.

Answer: D

64 candidates are competing for 5 positions at a consulting

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64 candidates are competing for 5 positions at a consulting

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615 to 715 in 15 Days (Ria's Strategies)

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