GMATinsight wrote:
In how many ways can 6 chocolates be distributed among 3 children?
A child may get any number of chocolates from 0 to 6 and all the chocolates are identical.A) 21
B) 28
C) 56
D) 112
E) 224
SOURCE: http://www.GMATinsight.comIt may seem odd at first, but this question can be reduced to the analogous question:
In how many ways can we arrange the letters in IIOOOOOO?Let me explain the relationship between this question and the original question.
First, however, let's say the three children are
child A, child B, and child COne possible arrangement of the 8 letters is
OOIOIOOOThis arrangement represents child A receiving
2 chocolates, child B receiving
1 chocolate, and child C receiving
3 chocolates
Likewise, the arrangement
OIOOOOIO represents child A receiving
1 chocolate, child B receiving
4 chocolates, and child C receiving
1 chocolate
And the arrangement
OOIIOOOO represents child A receiving
2 chocolates, child B receiving 0 chocolates, and child C receiving
4 chocolates
Okay, enough examples.
Now that we understand the relationship between the original question and the analogous question, let's see how many ways we can arrange the letters in IIOOOOOO
----ASIDE--------------------
When we want to arrange a group of items in which some of the items are identical, we can use something called the MISSISSIPPI rule. It goes like this:
If there are n objects where A of them are alike, another B of them are alike, another C of them are alike, and so on, then the total number of possible arrangements = n!/[(A!)(B!)(C!)....] So, for example, we can calculate the number of arrangements of the letters in MISSISSIPPI as follows:
There are
11 letters in total
There are
4 identical I's
There are
4 identical S's
There are
2 identical P's
So, the total number of possible arrangements =
11!/[(
4!)(
4!)(
2!)]
-------------------------------
In the case of the letters in IIOOOOOO, . . .
There are
8 letters in total
There are
2 identical I's
There are
6 identical O's
So, the total number of arrangements =
8!/(
2!)(
6!) = 28
Answer: B
Cheers,
Brent
_________________
Brent Hanneson – Creator of gmatprepnow.com
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