combinations question pls help!! (2 Viewers)

[kittyc4tkittycar]

Six teachers are seated around a round table to implement a program for the incoming Year 7 students. In how many ways can the seating be arranged if the more experienced teachers (Hattie, Patricia, and Melissa) must be seated separately so hey inspire the other three teachers?

The answer should be 3x2x2=12, but I don't understand why.
Thanks!

 

[pilot_djeddie]

Aight I got you:

I did this in a drawing because I must admit, even as a person who loves maths, permutations still really confuse me so, I hope this is self-explanatory.

In essence, you want to first treat the "groups" as one unit that can be swapped around (3!) and subsequently, account for the internal swapping which can occur within each group of two people (2!).

Again, really sorry but the logic is quite hard to explain. It's really "the vibe" that makes it make sense lol

 

[kittyc4tkittycar]

I understand the groups part, but for the internal swapping, it would happen 2! times, but isn't it for each of the three groups? So then, would it be 3!x2!x3?=36 ?

 

[pilot_djeddie]

I understand the groups part, but for the internal swapping, it would happen 2! times, but isn't it for each of the three groups? So then, would it be 3!x2!x3?=36 ?

Ah the reason why is because we go macro then micro. So we already focused on the three groups, then we say “within each group, there are 2!” ways of doing it. And since we already multiply it by 3!, thereby it’s just 3!*2!