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

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Joined: 08 Jul 2010
Posts: 2537
Location: India
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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, 07:52
2
2
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Difficulty:

75% (hard)

Question Stats:

37% (01:21) correct 63% (01:38) wrong based on 110 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, 08:18
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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Joined: 02 Sep 2009
Posts: 49915
Re: In how many ways can 6 chocolates be distributed among 3 children? A c  [#permalink]

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16 May 2017, 08:50
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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Re: In how many ways can 6 chocolates be distributed among 3 children? A c  [#permalink]

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09 Jan 2018, 11:19
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
Intern
Joined: 20 Nov 2017
Posts: 4
Re: In how many ways can 6 chocolates be distributed among 3 children? A c  [#permalink]

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26 Apr 2018, 09:02
why the formula of identical objects doen'st work here ?
Is there any easy way to solve such problems please ?
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Joined: 22 Aug 2013
Posts: 1348
Location: India
Re: In how many ways can 6 chocolates be distributed among 3 children? A c  [#permalink]

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26 Apr 2018, 11:23
2
1
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
Re: In how many ways can 6 chocolates be distributed among 3 children? A c &nbs [#permalink] 26 Apr 2018, 11:23
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