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

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

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New post 16 May 2017, 07:52
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Question Stats:

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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

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

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New post 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 \(y_i = x_i - 1\).

\(x_i = y_i +1\):

\(y_1 + y_2 + y_3 = 3\)

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

Answer 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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New post 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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New post 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
now we are in business

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

5!
_____
3! 2!

= 10, the answer

hope this helps
thanks

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

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

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New post 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
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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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