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# In how many ways can 12 different books be distributed equally among 4

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In how many ways can 12 different books be distributed equally among 4  [#permalink]

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24 Nov 2018, 10:03
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Difficulty:

75% (hard)

Question Stats:

37% (01:16) correct 63% (01:36) wrong based on 63 sessions

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GMATbuster's Weekly Quant Quiz#10 Ques #2

In how many ways can 12 different books be distributed equally among 4 different boxes?

A) 12C3
B) 12C4
C) 12C3*9C3*6C3
D) 12C4*8C4
E) 12C3*9C3*6C3*4!

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Re: In how many ways can 12 different books be distributed equally among 4  [#permalink]

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24 Nov 2018, 10:15
1
Books have to be divided equally. So each box should get 12/4 = 3 books only.
So 1st box gets 3 books from 12 in 12C3 ways. Now, 9 books remaining.
So 2nd box gets 3 books from 9 in 9C3 ways. Now 6 books remaining.
So 3rd box gets 3 books from 6 in 6C3 ways. Now 3 books remaining.
So 4th box gets 3 books from 3 left in 3C3 ways or 1 way.

Ways of distributing 12 books among 4 similar boxes will be 12C3*9C3*6C3*1
Assuming that different boxes means which books go into which box will make a difference, then we will be able to arrange the total ways of distributing 12 books in 4! ways.

So total ways = 12C3*9C3*6C3*1*4!

Option E
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Re: In how many ways can 12 different books be distributed equally among 4  [#permalink]

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24 Nov 2018, 10:27
E.
12 different books are there...and we have to divide in 4 boxes. So each group will have 3 books.

So number of ways for selecting 3 books is 12C3*9C3*6C3.

also since the 4 boxes are different we can arrange the group's among them.
So for that number of ways will be four.

So option E

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Re: In how many ways can 12 different books be distributed equally among 4  [#permalink]

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24 Nov 2018, 10:30
Person 1:
3 books out of 12= 12C3
Person 2:
3 books out of 9= 9C3
Person 3:
3 books out of 6= 6C3
Person 4:
3 books out of 3= 3C3

Thus total possible solutions:
12C3x9C3x6C3x3C3x4!

4! is used because the 4 people can be arranged in 4! ways.

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Re: In how many ways can 12 different books be distributed equally among 4  [#permalink]

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24 Nov 2018, 11:08
[quote="gmatbusters"]

GMATbuster&#39;s Weekly Quant Quiz#10 Ques #2

In how many ways can 12 different books be distributed equally among 4 different boxes?

A) 12C3
B) 12C4
C) 12C3*9C3*6C3
D) 12C4*8C4
E) 12C3*9C3*6C3*4!

12 different books be distributed equally among 4 different boxed in 12C3*9C3*6C3 ways
Ways to arrange these books in 4 different boxes is 4!

Sent from my iPhone using GMAT Club Forum mobile app
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Posts: 103
Re: In how many ways can 12 different books be distributed equally among 4  [#permalink]

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24 Nov 2018, 14:54
For distributing 12 books equally across 4 boxes, we need to split 3 books in each.

First 3 books can be selected in 12C3 ways, second 3 is 9C3 ways, third 3 in 6C3 ways and last 3 in 3C3 ways.

Now, let four boxes be A, B, C and D. First box can be selected in 4 ways as it can be any of A, B, C or D. Lets say A is selected as first box. Similarly, Second box can be selected in 3 ways as it can be any from B, C and D and third box can be selected in 2 ways and last box in 1 way.

So, total number of ways = 12C3*9C3*6C3*3C3*4*3*2*1 = 12C3*9C3*6C3*1*4!

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Posts: 4134
Re: In how many ways can 12 different books be distributed equally among 4  [#permalink]

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29 Jun 2019, 05:45
Top Contributor
gmatbusters wrote:

GMATbuster's Weekly Quant Quiz#10 Ques #2

In how many ways can 12 different books be distributed equally among 4 different boxes?

A) 12C3
B) 12C4
C) 12C3*9C3*6C3
D) 12C4*8C4
E) 12C3*9C3*6C3*4!

Take the task of distributing the books and break it into stages.

We must place 3 books in each of the 4 boxes. So, let's call the boxes box #1, box #2, box #3 and box #4

Stage 1: Select 3 books to go in box #1
Since the order in which we select the books does not matter, we can use combinations.
We can select 3 books from 12 books in 12C3 ways
So, we can complete stage 1 in 12C3 ways

Stage 2: Select 3 books to go in box #2
There are 9 boxes remaining.
So, we can complete this stage in 9C3 ways

Stage 3: Select 3 books to go in box #3
There are 6 boxes remaining.
So, we can complete this stage in 6C3 ways

Stage 4: Select 3 books to go in box #4
There are 3 boxes remaining.
So, we can complete this stage in 3C3 ways

By the Fundamental Counting Principle (FCP), we can complete all 4 stages (and thus distribute all 12 books) in (12C3)(9C3)(6C3)(3C3) ways

However, if we recognize that 3C3 = 1, we can see that [color=blue(12C3)(9C3)(6C3)(3C3)[/color] = (12C3)(9C3)(6C3)(1) = (12C3)(9C3)(6C3)

Note: the FCP can be used to solve the MAJORITY of counting questions on the GMAT. So, be sure to learn it.

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Re: In how many ways can 12 different books be distributed equally among 4   [#permalink] 29 Jun 2019, 05:45
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