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itsworththepain
GMAT 1: 720 Q49 V38
Posts: 12
Updated on: 27 Jun 2014, 01:47
Originally posted by itsworththepain on 26 Jun 2014, 23:53.
Last edited by Bunuel on 27 Jun 2014, 01:47, edited 2 times in total.
Last edited by Bunuel on 27 Jun 2014, 01:47, edited 2 times in total.
Edited the question and added the OA.
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A
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
55%
(hard)
Question Stats:
55% (01:07) correct
45%
(01:34)
wrong
based on 1049
sessions
History
Date
Time
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In how many ways can 5 different fruits be distributed among four children? (Some children may get more than one fruit and some may get no fruits.)
(A) 4^5
(B) 5^4
(C) 5!
(D) 4!
(E) 4!*5!
(A) 4^5
(B) 5^4
(C) 5!
(D) 4!
(E) 4!*5!
Correct Answer is 1024 and I understand the reasoning behind that.
But the approach I followed was:
Say we keep the 5th fruit aside and distribute the other 4 fruits among the 4 children
No of ways to distribute 4 fruits among 4 children = 4!
Now for each of these 4! Combinations, 5th fruit can be distributed to any of the 4 children
i.e. 4 new combinations for each of the 4! combinations
No of ways to distribute 5th fruit = 4*4!
5th fruit can be selected in 5 different ways
Total combinations are 5*4*4! = 480
What’s wrong here? What is it that I am missing here? What are the other 1024-480 combinations that I am missing?
But the approach I followed was:
Say we keep the 5th fruit aside and distribute the other 4 fruits among the 4 children
No of ways to distribute 4 fruits among 4 children = 4!
Now for each of these 4! Combinations, 5th fruit can be distributed to any of the 4 children
i.e. 4 new combinations for each of the 4! combinations
No of ways to distribute 5th fruit = 4*4!
5th fruit can be selected in 5 different ways
Total combinations are 5*4*4! = 480
What’s wrong here? What is it that I am missing here? What are the other 1024-480 combinations that I am missing?
27 Jun 2014, 00:08
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sagarsameer
I believe you picked up this question from my blog post so you know how to correctly solve it. So I will focus on only pointing out the error you committed. To start off, how do you get "No of ways to distribute 4 fruits among 4 children = 4! "? You are not given that each child gets only one fruit. If each child were to receive only one fruit, then you could have said that the first fruit can be given away in 4 ways, 2nd fruit in 3 ways and so on to get 4*3*2*1.
Some children may get none so others may get 2 or 3 or 4. Each fruit can be given out in 4 ways so 4*4*4*4*4 = 1024
Mugdho
There are 5 diff fruits to be distributed among 4 children.
First person does not choose a fruit in 5 ways. That would have been true if each child were to pick exactly 1 fruit. But here a child may not get any fruit and another may get 2 or more.
The only thing we know is that all 5 fruits are distributed. So each fruit gets an owner. Fruit 1 can be given away in 4 ways (to any one of the 4 children). Fruit 2 can be again given away in 4 ways to any 1 of the 4 children and so on.
There is no constraint on how many fruits each child may get.
That is why we get 4*4*4*4*4.
All these cases with small variations with distinct and identical objects (total 16 cases) are covered in our Combinatorics module in detail here. You can check it out using the 3 day trial.
https://anaprep.com
