Chris, haven't you tried to put the three people together rather than NOT together ?
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"7 position possibilities * 6 arrrangements of the three people for each position possibility* 24 arrangements of the other 4 people = 1008 ways"
I am going to assume that is what you have tried to find. And I will look at your logic.
Yes there are 6 ways to seat the 3 that will remain together - in relation to each other.
So we will take one of these arrangements and seat these three people.
Noy you are right i believe, there are 4! = 24 arangements for the other 4
6*24=144 Yes so far I agree.
The you multiply this andwer by 7..... Why did you do that ?