AAPL wrote:Magoosh
In how many ways can 16 different gifts be divided among four children such that each child receives exactly four gifts?
A. \(16^4\)
B. \((4!)^4\)
C. \(\frac{16!}{(4!)^4}\)
D. \(\frac{16!}{4!}\)
E. \(4^{16}\)
OA C
Let's say the children are named A, B, C, and D
Stage 1: Select 4 gifts to give to child A
Since the order in which we select the 4 gifts does not matter, we can use combinations.
We can select 4 gifts from 16 gifts in 16C4 ways (= 16!/(4!)(12!))
So, we can complete stage 1 in
16!/(4!)(12!) ways
Stage 2: select 4 gifts to give to child B
There are now 12 gifts remaining
Since the order in which we select the 4 gifts does not matter, we can use combinations.
We can select 4 gifts from 12 gifts in 12C4 ways (= 12!/(4!)(8!))
So, we can complete stage 2 in
12!/(4!)(8!) ways
Stage 3: select 4 gifts to give to child C
There are now 8 gifts remaining
We can select 4 gifts from 8 gifts in 8C4 ways (= 8!/(4!)(4!))
So, we can complete stage 3 in
8!/(4!)(4!) ways
Stage 4: select 4 gifts to give to child D
There are now 4 gifts remaining
NOTE: There's only 1 way to select 4 gifts from 4 gifts, but if we want the answer to look like the official answer, let's do the following:
We can select 4 gifts from 4 gifts in 4C4 ways (= 4!/4!)
So, we can complete stage 4 in
4!/4! ways
By the Fundamental Counting Principle (FCP), we can complete all 4 stages (and thus distribute all 16 gifts) in
[16!/(4!)(12!)][12!/(4!)(8!)][8!/(4!)(4!)][4!/4!] ways
A BUNCH of terms cancel out to give us = 16!/(4!)�
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 our free video:
https://www.gmatprepnow.com/module/gmat-counting?id=775
Then you can try solving the following questions:
EASY
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https://www.beatthegmat.com/what-should- ... 67256.html
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https://www.beatthegmat.com/counting-pro ... 44302.html
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https://www.beatthegmat.com/picking-a-5- ... 73110.html
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https://www.beatthegmat.com/permutation- ... 57412.html
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https://www.beatthegmat.com/simple-one-t270061.html
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https://www.beatthegmat.com/mouse-pellets-t274303.html
MEDIUM
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https://www.beatthegmat.com/combinatoric ... 73194.html
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https://www.beatthegmat.com/arabian-hors ... 50703.html
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https://www.beatthegmat.com/sub-sets-pro ... 73337.html
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https://www.beatthegmat.com/combinatoric ... 73180.html
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https://www.beatthegmat.com/digits-numbers-t270127.html
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https://www.beatthegmat.com/doubt-on-sep ... 71047.html
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https://www.beatthegmat.com/combinatoric ... 67079.html
DIFFICULT
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https://www.beatthegmat.com/wonderful-p- ... 71001.html
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https://www.beatthegmat.com/ps-counting-t273659.html
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https://www.beatthegmat.com/permutation- ... 73915.html
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https://www.beatthegmat.com/please-solve ... 71499.html
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https://www.beatthegmat.com/no-two-ladie ... 75661.html
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https://www.beatthegmat.com/laniera-s-co ... 15764.html
Cheers,
Brent
