In how many different ways can a group of 12 boxes be divided into 3

Question

In how many different ways can a group of 12 boxes be divided into 3 sub-groups of 4 boxes each?

A. 5775
B. 7700
C. 9130
D. 34650
E. 184800

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Nidzo · Accepted Answer

adi33

Choosing boxes for the first sub-group:  There are 12 boxes and we need 4 so the number of ways of doing this will be  \frac{12!}{4!8!} = 495

Choosing boxes for the second sub-group:  As we have chosen 4 for the first group, now we have only 8 boxes to choose 4 from. So it will be  \frac{8!}{4!4!} = 70  ways.

Choosing boxes for the third sub-group:  Now we only have 4 boxes left and need to choose 4, which can only be done in a single way.

Now if which subgroup four of the boxes were in mattered, one would merely multiply the three values to get the answer. However, which sub-group a group of four boxes belongs to is not important here which means that we need to account for repetition by dividing by the factorial of the number sub-groups.

Just for clarity let's take four boxes (boxes 1, 2, 3 & 4) and group them together and call them collectively  A . Then boxes 5, 6, 7 & 8 are grouped together and called  B  and lastly 9, 10, 11 & 12 are grouped together as  C . Going  495*70*1  would take  A, B & C  slot and arrange them into each sub-group. In how many ways can  A, B & C  be arranged in 3 subgroups?  3! = 6 :

ABC
ACB
BAC
BCA
CAB
CBA

However, we are only concerned with the fact that the groups of  A, B & C  can be created and which sub-group they are in does not matter. So by multiplying the number of outcomes we will end up with 6 times more outcomes than we want. To remove the repetition we have to divide by  3! = 6 .

So the answer will be:  \frac{495*70*1}{6} = 5775

Answer A

Archit3110 · Answer

total possible ways to divide 12 boxes in 3 sub groups of 4 boxes each :

(12c4*8c4*4c4) / 3! = 5575

OPTION A

ThatDudeKnows · Answer

Here's a little trick that saves a few seconds on questions like this. If you start with a big group and divide it into smaller ones, you take the factorial of the starting group and divide the factorials of each of the smaller groups. In this case, we have  \frac{12!}{4!4!4!} . That gives us the number of ways if we could have, say, boxes ABCD in sub-group 1, EFGH in 2, IJKL in 3 and treat that as different from EFGH in 1, ABCD in 2, and IJKL in 3. But those are the same. So we need to divide by the number of ways there are to arrange the three sub-groups, and there are 6 ways to arrange three elements:

\frac{12*11*10*9*8*7*6*5}{6*4*3*2*4*3*2} = 11*5*3*7*5 = 11*25*21 = 275*21 = 5500+275 = 5775

Answer choice A.

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In how many different ways can a group of 12 boxes be divided into 3

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In how many different ways can a group of 12 boxes be divided into 3

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