A school teacher wants to divide the whole class into 3 groups with...

Hi..
The reason is that we have not named the group..
So if A,B are in group I, C,D are in group 2 and E,F are in group 3..
It will be same when A,B are in group I, C,D are in group 3 and E,F are in group 2..
OR if A,B are in group 2, C,D are in group 1, and E,F are in group 3..

Since here all groups name and size wise are identical, all above distribution are identical, and hence we divide by 3!

­When you have reptitions that have been counted multiple times then you require to remove them by dividing them by number that each combination is repeated.
Say, you have to select two groups consisting of equal members from A, B, C and D.
4C2*2C2 = 6 ways, but do we have 6 ways.
(1,2) = (AB,CD), (AC,BD), (AD,BC), 
So, where are the remaining 3 ways, they are reverse of the above, that is (CD,AB), (BD,AC) and (BC,AD).
And, why did this happen? Because we took 1 and 2 group to be different, but they are identical.
Thus divide by ways these groups can be arranged 2! for 2 groups and 3! for 3 groups or n! for n groups.

Now, when do you multiply by 3!?
arranging 3 out of 5 on three chairs.
You select three person from five in 5C3 ways, and arrange them in 3! ways within themselves, so multiply by 3! = 5C3*3!

\(\frac{24!}{(8!)^3 * 3!} = \frac{_{24}C_8*_{16}C_8*_8C_8}{3!}\)

Answer B

chetan2u

Hi Chetan2u,

We are referring the groups identical because the no. of students are same?

YES NITISH..

since the groups have same number and there is no other difference, these groups are identical..

example ..
there are three bags- same colour and same size- all bags are identical..
but say the bags are different in colour. not identical.. we will not divide by 31 or so

Hi,
I understood till you came up to 24C8*16C8*8C8.. But, I don't know why are dividing by 3!.
We don't need to arrange the 3 groups right?, we just want to find how many ways to get 3 groups.
Lets take A,B,C are 3 groups.
We don't want to find ABC, ACB,CAB etc..

Clearly I am not following you. Can you please explain me

Thanks,

chetan2u
I don't understand why the groups are identical.The students are not identical so how can the groups be identical?
In such questions do we have to presume the groups as same identity and ignore the diversity of students?

Correct if wrong:

The way I see it, there are two steps to this.

1) e.g. think about if we have 6 children divided into 3 groups. 2 kids in each.

How many kids in the first group? 6C2 - you have 6 kids and you choose 2 of them to be in the first group.
How many kids in the second group? 4C2 - You already picked 2 kids to be in the first so 4 are remaining. choose 2 more out of 4.
How many in the third? 2C2 - Only 2 kids left and choosing both of them.

So, we have 6C2*4C2*2C2. BUT...

2) Now, you have these groups (lets say, A,B,C). You can ofcourse arrange them in the order ABC. The question is, is the order CBA different from ABC? The answer is no. They all have 2 kids in them and even though the kids in them are different people, their combinations have already been accounted for in step 1. SO really, A=B=C. Ie. How many ways can you arrange AAA? 3*2*1/3!=1 right?

Same concept in this situation - 3 groups all identical and we found there are 6C2*4C2*2C2 ways of creating them so divide by 3! to get the total number of ways. Now you can apply the same to the problem above.

Thoughts?

I answered as option E.

Please can anyone tell for these kind of questions... when should we divide by 3! and when should we multiply by 3!

understood.. thanks