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# In how many ways can 16 different gifts be divided among four children

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In how many ways can 16 different gifts be divided among four children [#permalink]

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26 Mar 2015, 03:25
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Question Stats:

61% (00:57) correct 39% (01:07) wrong based on 177 sessions

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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. 16!/(4!)^4
D. 16!/4!
E. 4^16

Kudos for a correct solution.
[Reveal] Spoiler: OA

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In how many ways can 16 different gifts be divided among four children [#permalink]

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26 Mar 2015, 05:09
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Bunuel wrote:
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. 16!/(4!)^4
D. 16!/4!
E. 4^16

Kudos for a correct solution.

$$C^4_1_6 * C^4_1_2 * C^4_8 = \frac{16 * 15 *14 *13}{4*3*2} * \frac{12*11*10*9}{4*3*2} * \frac{8*7*6*5}{4*3*2}$$
= $$\frac{16!}{4!^4}$$
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Re: In how many ways can 16 different gifts be divided among four children [#permalink]

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30 Mar 2015, 02:33
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Bunuel wrote:
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. 16!/(4!)^4
D. 16!/4!
E. 4^16

Kudos for a correct solution.

MAGOOSH OFFICIAL SOLUTION:
Attachment:

Count_Prob_16gifts.png [ 26.52 KiB | Viewed 4153 times ]

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Re: In how many ways can 16 different gifts be divided among four children [#permalink]

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30 Mar 2015, 09:46
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Total 16 different Gifts, and 4 children.
Thus any one child gets 16C4 gifts,
then the other child gets 12C4 gifts(16 total - 4 already given),
then the third one gets 8C4 gifts,
and the last child gets 4C4 gifts.
Since order in which each child gets the gift is not imp, thus, ans :
16C4 * 12C4 * 8C4 * 4C4 = 16! / (4!)^4
Ans : C.
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Re: In how many ways can 16 different gifts be divided among four children [#permalink]

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31 Mar 2015, 02:14
Bunuel wrote:
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. 16!/(4!)^4
D. 16!/4!
E. 4^16

Kudos for a correct solution.

16 gifts can be distributed to 4 children in 16C4 ways.
Remaining 12 gifts can be distributed to 4 children in 12C4 ways.
Remaining 8 gifts can be distributed to 4 children in 8C4 ways.
Lastly, remaining 4 gifts can be distributed to 4 children in 4C4 ways.

Total ways = 16C4 * 12C4 * 8C4 * 4C4
= 16!/(4!)^4

Hence option (C).
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Re: In how many ways can 16 different gifts be divided among four children [#permalink]

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04 Oct 2017, 10:24
Bunuel wrote:
Bunuel wrote:
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. 16!/(4!)^4
D. 16!/4!
E. 4^16

Kudos for a correct solution.

MAGOOSH OFFICIAL SOLUTION:
Attachment:

Don't we have to multiply by 4! because every child could get a different set of 4 gifts?

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Re: In how many ways can 16 different gifts be divided among four children   [#permalink] 04 Oct 2017, 10:24
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