Problem Solving

vaibhav101 wrote:in how many ways can 12 books be divided among 3 boys so that each receives 4 books?

A 36540
B 34560
C 34650
D 35640
E 36450

Take the task of distributing the 12 books and break it into stages

Let's say the children are named A, B and C

Stage 1: Select 4 books to give to child A
Since the order in which we select the 4 books does not matter, we can use combinations.
We can select 4 books from 12 books in 12C4 ways (= 12!/(4!)(8!))
So, we can complete stage 1 in 12!/(4!)(8!) ways

Stage 2: select 4 books to give to child B
There are now 8 books remaining
We can select 4 books from 8 books in 8C4 ways (= 8!/(4!)(4!))
So, we can complete stage 2 in 8!/(4!)(4!) ways

Stage 4: select 4 books to give to child C
There are now 4 books remaining
NOTE: There's only 1 way to select 4 books from 4 books, but if we want this result to look like the others, let's do the following:
We can select 4 books from 4 books in 4C4 ways (= 4!/4!)
So, we can complete stage 3 in 4!/4! ways

By the Fundamental Counting Principle (FCP), we can complete all 3 stages (and thus distribute all 12 books) in [12!/(4!)(8!)][8!/(4!)(4!)][4!/4!] ways

A BUNCH of terms cancel out to give us 12!/(4!)³, which evaluates to be 34650

Answer: C


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

Then you can try solving the following questions:

EASY
- https://www.beatthegmat.com/what-should ... 67256.html
- https://www.beatthegmat.com/counting-pr ... 44302.html
- https://www.beatthegmat.com/picking-a-5 ... 73110.html
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- https://www.beatthegmat.com/simple-one-t270061.html
- https://www.beatthegmat.com/mouse-pellets-t274303.html


MEDIUM
- https://www.beatthegmat.com/combinatori ... 73194.html
- https://www.beatthegmat.com/arabian-hor ... 50703.html
- https://www.beatthegmat.com/sub-sets-pr ... 73337.html
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- https://www.beatthegmat.com/digits-numbers-t270127.html
- https://www.beatthegmat.com/doubt-on-se ... 71047.html
- https://www.beatthegmat.com/combinatori ... 67079.html


DIFFICULT
- https://www.beatthegmat.com/wonderful-p ... 71001.html
- https://www.beatthegmat.com/ps-counting-t273659.html
- https://www.beatthegmat.com/permutation ... 73915.html
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- https://www.beatthegmat.com/laniera-s-c ... 15764.html

Cheers,
Brent

vaibhav101 wrote:in how many ways can 12 books be divided among 3 boys so that each receives 4 books?

A 36540
B 34560
C 34650
D 35640
E 36450

Once a boy receives 4 books, the order in which he receives them doesn't matter. So the first boy has 12C4 ways to receive his 4 books, the second boy has 8C4 ways to receive his 4 books and the third boy has 4C4 ways to receive his books. So the total number of ways the 3 boys can receive 4 books each is

12C4 x 8C4 x 4C4

(12 x 11 x 10 x 9)/(4 x 3 x 2) x (8 x 7 x 6 x 5)/(4 x 3 x 2) x 1

11 x 5 x 9 x 2 x 7 x 5

34650

Answer: C