In how many ways can 6 people be arranged in a round table if a certain 3
persons refuse to follow each other? 

I assume that if 3 persons refuse to follow each other, that means that no two
of the three can sit next to each other.  Then they can only be seated in this
general way:


           #
       @       @  

       #       #
           @


The three, no two of whom will sit together, can sit where the @'s are in 3!=6
ways and the other three can sit where the #'s are in 3!=6 ways.

Answer (3!)(3!) = (6)(6) = 36 ways

Edwin