Problem Solving

6! is your upper constraint.
each time one of those three special people is in their respective evil seat, there are 5! ways to arrange the other five people. But you can't eliminate 3 x 5! because there are repeats in that amount i.e. that counts when Tom and Jerry are in their evil seats multiple times. For example, when Jerry is in his evil seat, there are times out of those 5! arrangements when Tom happens to be in his evil seat and vice versa so you would overcount the arrangements. So you need to figure out the number of repeats out of that 3x5!. You reduce that by the number of times two of them are in their evil seats 3!/2! ways times 4! arrangements for 4 other people. Then, by the same logic, that number is reduced by the number of times that three of them are in their evil seats i.e. 1 way times 3! arrangements for the 3 other people. And then all the overcounting is accounted for. So the equation looks like:

=6! - ((3!/2!)*5! - ((3!/2!)*4! - (3!/3!)*3!))

=720-(3*120-(3*24-1*6))
=720-(360-66)
=720-294
=426

8) SNAP!

Ron, wow! excellent explanation. great help!

thanks, cramya