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# In how many ways can 6 chocolates be distributed among 3 children? A c

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GMAT Club Legend
Status: GMATINSIGHT Tutor
Joined: 08 Jul 2010
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In how many ways can 6 chocolates be distributed among 3 children? A c  [#permalink]

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16 May 2017, 06:52
2
16
00:00

Difficulty:

75% (hard)

Question Stats:

46% (01:31) correct 54% (02:00) wrong based on 185 sessions

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In how many ways can 6 chocolates be distributed among 3 children? A child may get any number of chocolates from 1 to 6 and all the chocolates are identical.

A) 10
B) 15
C) 21
D) 28
E) 56

SOURCE: http://www.GMATinsight.com

_________________
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Re: In how many ways can 6 chocolates be distributed among 3 children? A c  [#permalink]

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16 May 2017, 07:50
GMATinsight wrote:
In how many ways can 6 chocolates be distributed among 3 children? A child may get any number of chocolates from 1 to 6 and all the chocolates are identical.

A) 10
B) 15
C) 21
D) 28
E) 56

SOURCE: http://www.GMATinsight.com

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HARD (FOR PRACTICE): https://gmatclub.com/forum/5-rings-on-4 ... 86111.html
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Re: In how many ways can 6 chocolates be distributed among 3 children? A c  [#permalink]

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26 Apr 2018, 10:23
4
5
Awamy wrote:
why the formula of identical objects doen'st work here ?
Is there any easy way to solve such problems please ?

Hello

The formula of identical objects DOES work here. If we have to distribute N identical objects among R distinct groups such that one or more people might get none of the objects, then the formula is = (N+R-1) C (R-1) (selecting R-1 objects out of N+R-1 objects)

BUT if N identical objects have to be distributed among R distinct groups such that everyone should get at least one object, then the formula is = (N-1) C (R-1) (selecting R-1 objects out of N-1 objects)

So in this question, since 6 identical chocolates have to be distributed among 3 distinct people, but everyone should get at least one, we will apply the second formula = (6-1) C (3-1) = 5C2 = 10, which is our answer
##### General Discussion
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Re: In how many ways can 6 chocolates be distributed among 3 children? A c  [#permalink]

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16 May 2017, 07:18
1
1
GMATinsight wrote:
In how many ways can 6 chocolates be distributed among 3 children? A child may get any number of chocolates from 1 to 6 and all the chocolates are identical.

A) 10
B) 15
C) 21
D) 28
E) 56

SOURCE: http://www.GMATinsight.com

Hi

# of chocolates distributed to each child -$$x_1, x_2, x_3$$, where $$x_i > 0$$

We have non-empty set:

$$x_1 + x_2 + x_3 = 6$$

We need to convert it into $$x_i >=0$$ substituting each $$x_i$$ with $$y_i = x_i - 1$$.

$$x_i = y_i +1$$:

$$y_1 + y_2 + y_3 = 3$$

$$_{3+3-1}C_3 = _5C_3 = \frac{5*4}{2} = 10$$

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Re: In how many ways can 6 chocolates be distributed among 3 children? A c  [#permalink]

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09 Jan 2018, 10:19
3
1
GMATinsight wrote:
In how many ways can 6 chocolates be distributed among 3 children? A child may get any number of chocolates from 1 to 6 and all the chocolates are identical.

A) 10
B) 15
C) 21
D) 28
E) 56

SOURCE: http://www.GMATinsight.com
\\

since a child must get at least 1 chocolate, lets distribute 1 chocolate to each child first, and thus we are left with 3 chocolates to redistribute

since chocolates are identical, the remaining 3 chocolates can be distributed among 3 children as follows

5!
_____
3! 2!

hope this helps
thanks

cheers, and do consider some kudos, guys
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Re: In how many ways can 6 chocolates be distributed among 3 children? A c  [#permalink]

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26 Apr 2018, 08:02
why the formula of identical objects doen'st work here ?
Is there any easy way to solve such problems please ?
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Re: In how many ways can 6 chocolates be distributed among 3 children? A c  [#permalink]

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17 Dec 2018, 05:21
chetan2u Bunuel could you please explain how to solve this without (n-1)Cr-1 ... i wanna learn the concept
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Re: In how many ways can 6 chocolates be distributed among 3 children? A c  [#permalink]

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28 Jul 2019, 01:58
GMATinsight wrote:
In how many ways can 6 chocolates be distributed among 3 children? A child may get any number of chocolates from 1 to 6 and all the chocolates are identical.

A) 10
B) 15
C) 21
D) 28
E) 56

SOURCE: http://www.GMATinsight.com

formula to use here we can take case that child may get 0 chocolate
n-1Cr-1 ; n=6 . r= 3
5c2; 10
IMO A
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Re: In how many ways can 6 chocolates be distributed among 3 children? A c  [#permalink]

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26 Aug 2019, 13:02
GMATinsight wrote:
In how many ways can 6 chocolates be distributed among 3 children? A child may get any number of chocolates from 1 to 6 and all the chocolates are identical.

A) 10
B) 15
C) 21
D) 28
E) 56

SOURCE: http://www.GMATinsight.com

given: 6 identical chocs, 3 different kids, at least 1 each;

$$k_1+k_2+k_3=6…(k_1'+1)+(k_2+1)+(k_3+1)=6…k_1'+k_2+k_3=3$$
$$C(n+r-1,r-1)=(3+3-1,3-1)=\frac{5!}{2!3!}=10$$

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Re: In how many ways can 6 chocolates be distributed among 3 children? A c  [#permalink]

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28 Feb 2020, 23:10
GMATinsight wrote:
In how many ways can 6 chocolates be distributed among 3 children? A child may get any number of chocolates from 1 to 6 and all the chocolates are identical.

A) 10
B) 15
C) 21
D) 28
E) 56

SOURCE: http://www.GMATinsight.com

The detailed solution to the above problem using two methods is explained in the attached video

_________________
Prepare with PERFECTION to claim Q≥50 and V≥40 !!!
GMATinsight .............(Bhoopendra Singh and Dr.Sushma Jha)
e-mail: info@GMATinsight.com l Call : +91-9999687183 / 9891333772
One-on-One Skype classes l Classroom Coaching l On-demand Quant course l Admissions Consulting

Most affordable l Comprehensive l 2000+ Qn ALL with Video explanations l LINK: Courses and Pricing
Our SUCCESS STORIES: From 620 to 760 l Q-42 to Q-49 in 40 days l 590 to 710 + Wharton l
FREE GMAT Resource: 22 FREE (FULL LENGTH) GMAT CATs LINKS l NEW OG QUANT 50 Qn+VIDEO Sol.
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Re: In how many ways can 6 chocolates be distributed among 3 children? A c  [#permalink]

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17 Apr 2020, 10:10
GMATinsight wrote:
In how many ways can 6 chocolates be distributed among 3 children? A child may get any number of chocolates from 1 to 6 and all the chocolates are identical.

A) 10
B) 15
C) 21
D) 28
E) 56

SOURCE: http://www.GMATinsight.com

Let the children be A, B, and C. So A can get 1, B can get 1 and C can get 4 chocolates. Of course, this is different from A gets 4, B 1, and C 1, or, A gets 1, B 4, and C 1.

In the calculations below, we will show how 3 positive integers can sum to 6 and the number of ways the 3 numbers can be rearranged among A, B, and C (for example, the first calculation below describes the distribution of the 6 chocolates mentioned above):

1 + 1 + 4 = 6

3!/2! = 3 ways

1 + 2 + 3 = 6

3! = 6 ways

2 + 2 + 2 = 6

3!/3! = 1 way

Therefore, there are a total of 3 + 6 + 1 = 10 ways that 6 chocolates can be distributed to 3 children.

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Re: In how many ways can 6 chocolates be distributed among 3 children? A c   [#permalink] 17 Apr 2020, 10:10