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

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BTGModeratorVI: Legendary Member; Posts: 1223; Joined: Sat Feb 15, 2020 2:23 pmFollowed by: 1 member

by BTGModeratorVI » Wed Apr 22, 2020 11:06 am

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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!

Answer: C
Source: E-gmat

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Jay@ManhattanReview: GMAT Instructor; Posts: 3008; Joined: Mon Aug 22, 2016 6:19 amFollowed by: 34 members; Location: Grand Central / New York; Thanked: 470 times

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

by Jay@ManhattanReview » Thu Apr 23, 2020 10:25 pm

BTGModeratorVI wrote: ↑
Wed Apr 22, 2020 11:06 am
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!

Answer: C
Source: E-gmat

So, the 12 distinct books are to be equally distributed among 4 different boxes.

So each box will have 12/4 = 3 books

Three books can be distributed among 4 boxes in the following way...

• The 1st box will get 3 books from 12 books in 12C3 ways.
• The 2nd box will get 3 books from the remaining 9 books in 9C3 ways.
• The 3rd box will get 3 books from the remaining 6 books in 6C3 ways.
• The 4th box will get 3 books from the remaining 3 books in 3C3 ways i.e in only 1 way.

Total no. of ways = 12C3*9C3*6C3*1 = 12C3*9C3*6C3

The correct answer: C

Hope this helps!

-Jay
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deloitte247: Legendary Member; Posts: 2214; Joined: Fri Mar 02, 2018 2:22 pmFollowed by: 5 members

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

by deloitte247 » Fri Apr 24, 2020 7:32 am

In how many ways can 12 different books be distributed equally among 4 different boxes?
In order to distribute equally 12/4 = 3
i.e 3 books must be placed in each of the 4 boxes.

In selecting 3 books for the first box out of the available 12 books = 12C3
After selecting 3 books for box 1, books remain 12 - 3 = 9

In selecting 3 books for the second box out of the remaining 9 = 9C3
After selecting 3 books for box 2, books remain 9 - 3 = 6

In selecting 3 books for the third box out of the remaining 6 = 6C3
After selecting 3 books for box 3, books remain 6 - 3 = 3

In selecting 3 books for the fourth box out of the remaining 3 = 3C3
And then the books would have been equally distributed

Total number of ways to select and distribute the books = (12C3) * (9C3) * (6C3) * (3C3)
$$Since\ 3C3\ =\ \frac{3!}{3!\left(3-3\right)!}=\frac{3\cdot2\cdot1}{\left(3\cdot2\cdot1\right)\left(0!\right)}=\frac{6}{6\cdot1}=\frac{6}{6}=\ 1$$
Number of ways = (12C3)*(9C3)*(6C3)

Answer = C

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

by Brent@GMATPrepNow » Fri Apr 24, 2020 7:48 am

BTGModeratorVI wrote: ↑
Wed Apr 22, 2020 11:06 am
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!

Answer: C
Source: E-gmat

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

Check the answer choices....our answer doesn't seem to be there.
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)

Answer: C

Note: the FCP can be used to solve the MAJORITY of counting questions on the GMAT. For more information about the FCP, watch this free video: https://www.gmatprepnow.com/module/gmat- ... /video/775

You can also watch a demonstration of the FCP in action: https://www.gmatprepnow.com/module/gmat ... /video/776

Then you can try solving the following questions:

EASY
- https://www.beatthegmat.com/what-should- ... 67256.html
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Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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Scott@TargetTestPrep: GMAT Instructor; Posts: 8176; Joined: Sat Apr 25, 2015 10:56 amFollowed by: 29 members; Location: Los Angeles, CA; Thanked: 43 times

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

by Scott@TargetTestPrep » Fri Apr 24, 2020 8:22 am

BTGModeratorVI wrote: ↑
Wed Apr 22, 2020 11:06 am
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!

Answer: C
Source: E-gmat

Each box will contain 3 books. There are 12C3 ways to put 3 books into the first box, 9C3 ways to put 3 books into the second box, 6C3 ways to put 3 books into the third box, and 3C3 ways to put the last 3 books into the fourth box. Therefore, the total number of ways is:

12C3 x 9C3 x 6C3 x 3C3 = 12C3 x 9C3 x 6C3

Answer: C

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