shankar245 wrote:

Can you please explain this?

I understand that we divide the slots! to remove identical stuff but here how does it make sense?

This is the logic behind this step:

Say there are 4 boys: A, B, C, D

There are two ways of splitting them in two groups.

Method I

The two groups can be made in the following ways

1. AB and CD

2. AC and BD

3. AD and BC

The groups are not named/distinct. You have 4 boys in front of you and you split them in 2 groups and do not name the groups. There are 3 total ways of doing this.

Method II

On the other hand, I could put them in two distinct groups in the following ways

1. Group1: AB, Group2: CD

2. Group1: CD, Group2: AB (If you notice, this is the same as above, just that now AB is group 2)

3. Group1: AC, Group2: BD

4. Group1: BD, Group2: AC

5. Group1: AD, Group2: BC

6. Group1: BC, Group2: AD

Here I have to put them in two different groups, group 1 and group 2. AB and CD is not just one way of splitting them. AB could be assigned to group 1 or group 2 so there are 2 cases. In this case, every 'way' we get above will have two possibilities so total number of ways will be twice.

So there will be 6 total ways.

Here since the groups are not distinct but 8C2 * 6C2 * 4C2 * 2C2 makes them distinct (we say, select the FIRST group in 8C2 ways, SECOND group in 6C2 ways etc), we need to divide by 4!.

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