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

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Math Expert
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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, 04:25
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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, 06:09
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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, 03:33
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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 5291 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, 10:46
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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, 03:14
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).
--
Optimus Prep's GMAT On Demand course for only $299 covers all verbal and quant. concepts in detail. Visit the following link to get your 7 days free trial account: http://www.optimus-prep.com/gmat-on-demand-course SVP Joined: 12 Dec 2016 Posts: 1945 Location: United States GMAT 1: 700 Q49 V33 GPA: 3.64 Re: In how many ways can 16 different gifts be divided among four children [#permalink] ### Show Tags 14 Jan 2018, 02:52 help, VeritasPrepKarishma why the formula (ax)! / a! (x!)^a does not work? Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 7988 Location: Pune, India Re: In how many ways can 16 different gifts be divided among four children [#permalink] ### Show Tags 16 Jan 2018, 01:04 chesstitans wrote: help, VeritasPrepKarishma why the formula (ax)! / a! (x!)^a does not work? Note that you are distributing 16 gifts among 4 children. The children are distinct. If you were distributing the gifts among 4 identical baskets such that each basket has exactly 4 gifts, then you would need to divide by 4! too. Note where this formula comes from: Put all 16 gifts in a row in 16! ways. G1, G2, G3, G4, ... , G16 Now split them into 4 groups G1, G2, G3, G4 || G5, G6, G7, G8 || ... || G13, G14, G15, G16 If the 4 groups are identical (like identical baskets), divide by 4! 16!/4! Now since we don't want to arrange the gifts within the group, divide by 4! four times 16!/4!*(4!)^4 But since we have 4 distinct children here, we will not divide by the first 4! _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

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

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18 Jan 2018, 14:23
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

The first child can choose any 4 gifts from the 16 gifts; thus (s)he has 16C4 ways to choose them. Once (s)he has chosen his or her 4 gifts, the second child can choose any 4 gifts from the remaining 12 gifts; thus (s)he has 12C4 ways to choose them. Likewise, the third child has 8C4 ways to choose his or her 4 gifts and the last child has 4C4 ways to choose his or her 4 gifts.
Thus the total number of ways the 16 gifts can be divided among the four children such that each child will receive 4 gifts is:

16C4 x 12C4 x 8C4 x 4C4

(16 x 15 x 14 x 13)/4! x (12 x 11 x 10 x 9)/4! x (8 x 7 x 6 x 5)/4! x (4 x 3 x 2 x 1)/4!

(16 x 15 x 14 x 13 x … x 4 x 3 x 2 x 1)/(4! x 4! x 4! x 4!)

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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21 Jan 2018, 00:35
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.

METHOD-1

Gift for first child can be selected in 16C4 ways = 1820
Gift for first child can be selected in 12C4 ways = 495
Gift for first child can be selected in 8C4 ways = 70
Gift for first child can be selected in 4C4 ways = 1

Total Ways to distribute gifts = (16C4*12C4*8C4*4C4) = 16!/(4!)^4

METHOD-2

We can arrange the 16 gifts in 16! ways considering that first 4 gifts are for 1st child, next 4 gifts are for 2nd child and so on...

But since the arrangement of 4 gifts received by child is irrelevant included in 16! so we need to eliminate the effect by dividing the result by 4! four times as there are four groups of 4gifts each group

hence answer = 16!/(4!^4)

In my opinion, the numbers given here are too big for GMAT to consider. This questions would have been apt if there were 4 children with 8 gifts where each child was to receive 2 gifts.
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Re: In how many ways can 16 different gifts be divided among four children   [#permalink] 21 Jan 2018, 00:35
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# In how many ways can 16 different gifts be divided among four children

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