Four persons enter a lift on the ground floor of an 8-floor building

Question

Four persons enter a lift on the ground floor of an 8-floor building(excluding the ground floor). Each person is equally likely to leave the lift on any floor except the ground floor. What is the probability of all the persons leaving the lift on different floors?
 
A. 105/512

B. 105/256

C. 112/243

D. 1560/2147

E. 7/8

GMATinsight · Accepted Answer

Out of 4 persons (Let, A, B, C and D) the first person has 8 choices (each at a floor) to leave the lift

The second person also has 8 choices to leave the lift

Likewise total possibilities of people to leave lift at floors = 8*8*8*8

Favourable ways to leave the lift
First person has 8 choices, second has 7 choices, third has 6 choices and forth has 5 choices so total ways = 8*7*6*5

Favourable Probability = (8*7*6*5) / (8*8*8*8) = 105 / 256

Answer: Option B

Archit3110 · Answer

total floors where 4 persons can get down ; 8^4
and since each person is likely to get down on different floor ; 8*7*6*5 
8*7*6*5 /8^4 = 105/256
IMO B

Mihiran · Answer

Can someone please tell me the error in my approach.

No. of ways 4 different floors can be selected out of 8 different floors=8C4=8!/(4!*4!)=70 reasoning: if 4 people leave on different floors,then 4 floors will be SELECTED.

No. of ways 1 floor can be SELECTED=8C1=8 reasoning:if all choose to leave at the same floor.

No. of ways 2 floors can be SELECTED=8C2=28 reasoning: if any combination of people decide to get out on 2 floors.

No. of ways 3 floors can be SELECTED=8C3=56 reasoning: if any combination of people decide to get out on 3 floors.

No. of ways 2 floors can be SELECTED=8C4=70 reasoning: if 4 people leave on different floors.
eqn: 8C4/(8C1+8C2+8C3+8C4)=35/81(wrong answer)

Sreeragc · Answer

i followed the reverse approach where i took 1-p(all leave on same floor) p(all leave on same floor)= 1/8 * 1/8 * 1/8 * 1/8 = 1/4096 so p(all leave on different floor) 1-(1/4096) = 4095/4096

where exactly i went wrong, cant we do it in this way? the answer seems to be wrong anyway!! Please help

hammypancakey · Answer

The first person can leave the lift on ANY floor. 
The second person can leave the lift on any floor EXCEPT the floor the first person left. 
The third person can leave the lift on any floor EXCEPT the floors the first and the second person left.
The last person can leave the lift on any floor EXCEPT the floors the first, the second, and the third person left.

Since the first person can leave on any floor, he/she has 8 out of 8 choices. 8/8=1
The second person has 7 out of 8 choices. 7/8.
The third person has 6 out of 8 choices. 6/8.
The fourth person has 5 out of 8 choices. 5/8.

1*(7/8)*(6/8)*(5/8) = 105/256

The correct answer is B.

Lionila · Answer

Dear Karishma,

Hope you are well.

My question relates to this forum post: https://gmatclub.com/forum/four-persons ... 90302.html

I do understand that the question can be solved with p=8*7*6*5/8^4

However, I am trying to solve it as well by calculating all possible cases for the denominator. Here is my approach.

1) all leave on the same floor: 8 ways

2)three leave on the same floor: 4C3 * 8 * 7

3) two and another two leave on the same floor: 4C2* 8 * 7

4) two leave on the same floor and the other two on different floors: 4C2 * 8 * 7* 6

5) all leave on different floors: 8*7*6*5

Calculating the probability:

(8*7*6*5 ) / (8*7*6*5 ) + (4C2 * 8 * 7* 6) + (4C2* 8 * 7 ) + (4C3 * 8 * 7 ) + 8 = 1680/4264 ( not the same as correct answer choice)

Do you see the mistake in this approach?

Thanks already!

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Four persons enter a lift on the ground floor of an 8-floor building

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Four persons enter a lift on the ground floor of an 8-floor building

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