SOLUTION: how many was can 6 people be seated in a row if 3 of them insist on NOT sitting together?

I can't tell whether you mean that it is OK if two
of them sit together as long as the third one isn't
next to either of them, or whether you mean that no 
two of the three can sit together.

I will assume that it is OK if just two of them sit
together, but not all three.

There are 6! or 720 ways any of the 6 can sit anywhere.

Now we must subtract all the ways in which the three
sit together.  

There are 3! was they can sit in three adjacent seats.

When they sit together it is the same as seating 4 things
in order, because the three boys together is like a single
thing. So we must subtract 3!*4! from the 6!. So the answer is 

6! - 3!*4! = 720 = 6*24 = 720 - 144 = 576

It's a more complicated problem if no two of the three can
sit together.  The answer to that one is 144.  I won't show you
that one unless that was what you meant.

Edwin